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# Cutkosky rules and 1-loop $\kappa$-deformed amplitudes

Andrea Bevilacqua  National Centre for Nuclear Research, ul. Pasteura 7,
02-093 Warsaw, Poland

###### Abstract

In this paper we show that the Cutkosky cutting rules are still valid term by
term in the expansion in powers of $\kappa$ of the $\kappa$-deformed 1-loop
correction to the propagator. We first present a general argument which
relates each term in the expansion to a non-deformed amplitude containing
additional propagators with mass $M>\kappa$. We then show the same thing more
pragmatically, by reducing the singularity structure of the coefficients in
the expansion of the $\kappa$-deformed amplitude, to the singularity structure
of non-deformed loop amplitudes, by using algebraic and analytic identities.
We will explicitly show this up to second order in $1/\kappa$, but the
technique can be generalized to higher orders in $1/\kappa$. Both the abstract
and the more direct approach easily generalize to different deformed theories.
We will then compute the full imaginary part of the $\kappa$-deformed 1-loop
correction to the propagator in a specific model, up to second order in the
expansion in $1/\kappa$, highlighting the usefulness of the approach for the
phenomenology of deformed models. This explicitly confirms previous
qualitative arguments concerning the behaviour of the decay width of unstable
particles in the considered model.

##  I Introduction

The unification of quantum mechanics (QM) and general relativity (GR) has
played a central role in theoretical physics for many decades (for a
historical panoramic see [1]). Both theories are very successful within their
respective domains, but they are based on very different physical assumptions,
which in turn manifest themselves into their mathematical description. The
unification of the two framework in a single theory of quantum gravity (QG),
therefore, poses significant challenges both from a physical point of view,
and from a more technical, mathematical point of view.

An interesting line of research originated from the pursuit of a final QG
theory is represented by potential effective low-energy theories of a more
fundamental QG model, such as those based on non-commutative spacetimes [2, 3,
4, 5, 6, 7, 8, 9, 10, 11]. An advantage of this approach over more direct ones
(in which QG models are built directly) is that it allows for an easier
contact with phenomenology, for example in the context of astrophysical
neutrinos [12, 13, 14, 15, 16]. In this paper we will focus on the
$\kappa$-Minkowsky spacetime, with commutation relations
$[x^{0},\mathbf{x}]=i\frac{\mathbf{x}}{\kappa}$ (for a review, see for example
[17, 18, 19] and references therein). In this model, the quantity $\kappa$ has
the dimension of mass, and it is usually identified with the Planck mass
$m_{p}$. As such, it is often sufficient to calculate the first order
correction in the small parameter (in our case proportional to $1/\kappa$) to
canonical, non-deformed quantities to obtain predictions of such deformed
models.

In previous publications [20, 26, 27], we developed and analyzed a model of a
(complex) scalar field in $\kappa$-Minkowski spacetime, with a particular
emphasis on its discrete symmetries — charge conjugation (C), parity (P), and
time reversal (T) — as well as the algebra of the associated Noether charges.
One interesting aspect of this model is obtained in the study of the decay
times of particles and antiparticles, in particular in different reference
frames [27, 23, 24]. Due to the nature of deformation, even non-interacting
models can exhibit non-trivial phenomenology. A key point in the investigation
of the decay of particles and antiparticles is the determination of higher
order corrections to the propagator of the decaying particle (and in
particular their imaginary part). Some preliminary results in this sense have
been obtained in [30], where (for simplicity) only the imaginary part coming
from the logarithms was calculated.

In particular, in this paper we focus on the Cutkosky cutting rules for
Feynman diagrams in order to compute the imaginary part of the loop
correction. In non-deformed QFT, they provide a pragmatic way of obtaining the
imaginary part of Feynman amplitudes (especially at 1-loop). In the
literature, the validity of the Cutkosky rules has already been investigated
for deformed models in $2+1$ dimensions [25], where it was possible to
integrate the full deformed amplitude explicitly over momentum space. More
general models (like the one considered here) lack this advantage. For the
sake of definiteness, we will consider the model presented in [20, 26, 27] and
obtain the $\kappa$-deformed amplitude for the 1-loop correction to the
propagator, expanded in powers of $1/\kappa$. One can then apply the Cutkosky
rule term by term in the series. We first present a general way to relate the
singularity structure of each term in the series, to the structure of a non-
deformed diagram with additional heavy particles. To actually proceed with the
computations, however, instead of applying the Cutkosky cutting rules
directly, which may be cumbersome in the deformed context since they involve
derivatives of Dirac deltas, we first use some analytic and algebraic
identities to rewrite the amplitude in terms of simpler ones. We will obtain
some ‘reduction identities’, which relate the singularity structure of each
term in the series in powers of $1/\kappa$ to non-deformed amplitudes, but in
a different way than what has been discussed above. These will then allow us
to use directly the Cutkosky rules, which can therefore be used term to term
in perturbation, and shows their validity in the expanded amplitude.

Our procedure is completely general. It can be applied to any theory with
deformed sum of momenta which reduce to the canonical sum in the non-deformed
limit, provided that the only singularities come from a deformation of the
momentum conservation in the propagator. Applying the procedure to the model
introduced in [20, 26, 27], after expanding the amplitude up to second order
in powers of $1/\kappa$, allows us to obtain phenomenological predictions.
This will allow us to explicitly obtain the first non-trivial correction due
to deformation of the decay width for unstable particles, confirming the
qualitative results obtained in [27]. In particular, we show that the
deformation correction to the decay width $\Gamma$ cannot be larger than
$p_{0}^{2}/\kappa^{2}$ in this model ($p_{0}$ is the energy of the decaying
particle), so that it can be neglected when extracting phenomenological
consequences of deformation, see for example [24, 27].

The paper is organized as follows. In section II we will briefly discuss the
Feynman propagator in the deformed context. In section III.1 we will then
quickly review the computation of the imaginary part of the 1-loop correction
to a scalar propagator (in a $\varphi\phi^{2}$ theory), both using the
canonical approach, and using the Cutkosky cutting rules. Then, in section IV
we will introduce our deformed amplitude for the 1-loop correction to the
propagator, and we will expand up to second order in powers of $1/\kappa$. The
Cutkosky rules are then discussed in section V. In this section, we relate the
singularity structure of each term in the expansion of the deformed amplitude,
to the singularity structure of a non-deformed diagram, containing propagators
with $M>\kappa$. We will then proceed to the more pragmatic approach in
section VI. In particular, we will obtain two reduction identities relating
the expansion of the deformed amplitude in powers of $1/\kappa$, to non-
deformed amplitudes. These relations are well defined provided that the
integrals are finite, which is the case since we are interested in the
imaginary part of the correction. Before moving to the actual computation of
the imaginary part of the amplitude in section IV, in section VI.1 we will
first provide an example of check of the validity of one of our reduction
relations obtained in section VI. We will then compute the imaginary part of
the $\kappa$-deformed 1-loop correction to the amplitude in section VII. For
simplicity, we will carry out the explicit computations in the centre of mass
(c.o.m.) frame, and we will then argue about the result in a general frame. We
will then discuss and summarize our results in section VIII.

##  II $\kappa$-deformed Feynman propagator

The model into consideration is the $\kappa$-deformed complex scalar field
introduced in [26]. In this model, the action is given by

$\displaystyle S_{\text{free}}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{4}}d^{4}x\,\,(\partial^{\mu}\phi)^{\dagger}\star(\partial_{\mu}\phi)+(\partial_{\mu}\phi)\star(\partial^{\mu}\phi)^{\dagger}-m^{2}\phi^{\dagger}\star\phi-m^{2}\phi\star\phi^{\dagger}$
(1)

and the field (and its Hermitian conjugate) satisfies the Klein-Gordon
equation.

In non-deformed QFT one can get the propagator in several equivalent ways, for
example by directly computing the VEV of the time ordered product of two
fields, as the Green function of the field operator, or by taking functional
derivatives of the path integral. In the deformed context, the first of these
approaches would require us to to compute objects like
$\phi^{\dagger}(x)\star\phi(y)$, which in turn implies that we need to make
some assumption for $[x^{0},\mathbf{y}^{i}]$. The second approach provides a
straightforward computation, since following [29] one can use the definitions

$\displaystyle
Z[J,J^{\dagger}]=\frac{1}{Z[0,0]}\int\mathscr{D}[\phi]\mathscr{[}\phi^{\dagger}]e^{iS_{\text{free}}[\phi,\phi^{\dagger}]+\frac{i}{2}\int[\phi^{\dagger}\star
J+J\star\phi^{\dagger}+J^{\dagger}\star\phi+\phi\star J^{\dagger}]}$ (2)

and the off-shell momentum-space action

$\displaystyle
S=\frac{1}{2}\int\frac{d^{4}p}{p_{4}^{2}/\kappa^{2}}\,\theta(p_{0})\,\zeta^{2}(p)\left(1+\frac{|p_{+}|^{3}}{\kappa^{3}}\right)(p_{\mu}p^{\mu}-m^{2})[a^{\dagger}_{\mathbf{p}}a_{\mathbf{p}}+b^{\dagger}_{\mathbf{p}^{*}}b_{\mathbf{p}^{*}}]$
(3)

to show that the propagator in momentum space is the same as the non-deformed
one [30]. However, as shown in [30], the above off-shell action can only be
defined in some subset of momentum space, since one needs to ensure that off-
shell one has $\omega_{p}\oplus\omega_{q}>0$ and $S(\omega_{p})\oplus
S(\omega_{q})<0$ (these inequalities are satisfied on-shell). For these
reasons, we will postpone the discussion of these two approaches to future
publications, and we concentrate on the propagator as the Green function of
the field operator. Since the field operator is not deformed, we simply have
that the propagator in momentum space is given by the canonical scalar
propagator.

##  III Example of loop corrections to the propagator: non-deformed case

###  III.1 Quick review of the direct computation

We consider the following diagram in a $\psi\phi^{2}$ theory, where both
$\psi$ and $\phi$ are scalars of mass $m_{\psi}$ and $m$ respectively.

With interaction vertex $g\psi\frac{\phi^{2}}{2!}$, the vertex is given by
$ig$ and the propagator is the canonical one. The amplitude can be written as
follows.

$\displaystyle
i\Pi(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{(p-k)^{2}-m^{2}+i\epsilon}.$
(4)

Using Feynman parameters, performing the Wick rotation, and using the identity

$\displaystyle\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{4}}\frac{(k^{2})^{a}}{(k^{2}+\Delta)^{b}}=\frac{\Gamma(b-a-\frac{1}{2}d)\Gamma(a+\frac{1}{2}d)}{(4\pi)^{d/2}\Gamma(b)\Gamma(\frac{1}{2}d)}\Delta^{-(b-a-d/2)}\mu^{4-d},$
(5)

the momentum space integral becomes

$\displaystyle\frac{\Gamma(\frac{\epsilon}{2})}{(4\pi)^{2}\Gamma(2)}\Delta^{-\frac{\epsilon}{2}}(\mu^{2})^{\frac{\epsilon}{2}}.$
(6)

Upon expanding, it reduces to

$\displaystyle\Pi(p^{2})=-\frac{(ig)^{2}}{(4\pi)^{2}}\int_{0}^{1}dx\,\left[\frac{2}{\epsilon}-\log\frac{\Delta}{\mu^{2}}-\gamma+O(\epsilon)\right].$
(7)

It is then sufficient to notice that the argument of the logarithm is positive
unless $p^{2}x(1-x)>m^{2}$, in which case the numerator becomes negative, and
therefore the whole argument of the logarithm becomes negative. Therefore, as
long as $L_{1}<x<L_{2}$ where
$L_{1}<\frac{1}{2}-\frac{1}{2}\sqrt{1-4\frac{m^{2}}{p^{2}}}$ and
$L_{2}<\frac{1}{2}+\frac{1}{2}\sqrt{1-4\frac{m^{2}}{p^{2}}}$, and if
$p^{2}>4m^{2}$, the amplitude gains a small imaginary part. In this case
$\Im\log\frac{\Delta}{|\Delta^{0}|}=-\pi$, and we finally have

$\displaystyle\Im\Pi(p^{2})=\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}.$
(8)

###  III.2 Quick review of the Cutkosky rules

The Cutkosky rules [31] allow one to compute the imaginary part of a diagram
by summing over all the possible cuts of the diagram, were for each cut
propagator one performs the substitution

$\displaystyle\frac{1}{D}\mapsto 2\pi\theta(p_{0})i\delta(D).$ (9)

Here we apply these rules to compute once again the imaginary part of the non-
deformed loop. From our initial definition

$\displaystyle
i\Pi(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\underbrace{\frac{i}{k^{2}-m^{2}+i\epsilon}}_{B}\underbrace{\frac{i}{(p-k)^{2}-m^{2}+i\epsilon}}_{A}$
(10)

there is only one possible cut, the one which cuts both propagators of the
loop. We get

$\displaystyle
2\Im[i\Pi(p^{2})]=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}q\,\,(2\pi
i)^{2}\theta(k_{0})\theta(q_{0})\delta(k^{2}-m^{2})\delta(q^{2}-m^{2})\delta^{4}(q-(p-k))$
(11)

which can be rewritten as

$\displaystyle
2\Im[i\Pi(p^{2})]=g^{2}\int\frac{d^{3}k}{(2\pi)^{2}}d^{3}q\,\,\frac{1}{4k_{0}q_{0}}\delta^{4}(q-(p-k))=\frac{g^{2}}{16\pi^{2}}\int
d^{3}k\frac{1}{k_{0}q_{0}}\delta(q_{0}+k_{0}-p_{0}).$ (12)

It is convenient to switch to the center of mass frame, in which
$p_{\mu}=(p_{0},0,0,0)$, and
$k_{\mu}=(\sqrt{|\mathbf{k}|^{2}+m^{2}},\mathbf{k})$ and
$q_{\mu}=(\sqrt{|\mathbf{k}|^{2}+m^{2}},-\mathbf{k})$ (recall that all the
particles are on-shell). The energies of the two particles after the vertex
are the same, and their sum is equal to the initial energy. This means that
the above integral becomes (we also pass to polar coordinates)

$\displaystyle 2\Im[i\Pi(p^{2})]$ $\displaystyle=\frac{g^{2}}{16\pi^{2}}\int
d|\mathbf{k}|\,4\pi|\mathbf{k}|^{2}\,\frac{1}{|\mathbf{k}|^{2}+m^{2}}\delta(2\sqrt{|\mathbf{k}|^{2}+m^{2}}-p_{0})$
(13) $\displaystyle=\frac{g^{2}}{8\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}$ (14)

where we used the fact that $p_{0}^{2}=p^{2}$ in the center of mass frame.
Therefore we get

$\displaystyle\Im[i\Pi(p^{2})]=\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}},$
(15)

which is of course the same result as before.

##  IV The $\kappa$-deformed amplitude

To compute the 1-loop correction in the $\kappa$-deformed context, we assume
that the vertex contribution remains $ig$, but the canonical momentum
conservation is switched with the deformed one.

We first consider the case of one incoming particle with momentum $S(p)$, and
two outgoing ones (let us call them $p_{2}$ and $p_{3}$). The conservation of
momentum then implies that

$\displaystyle S(p)\oplus p_{2}\oplus p_{3}=0$ (16)

This identity is immediately satisfied if $p_{2}\oplus p_{3}=p$, and we can
use one of the following parametrizations for $p_{2}$ and $p_{3}$

$\displaystyle\left\\{p_{2}=k,\,\,p_{3}=S(k)\oplus
p\right\\},\quad\left\\{p_{2}=p\oplus S(k),\,\,p_{3}=k\right\\}$ (17)
$\displaystyle\left\\{p_{2}=S(k),\,\,p_{3}=k\oplus
p\right\\},\quad\left\\{p_{2}=p\oplus k,\,\,p_{3}=S(k)\right\\}$ (18)

Notice that the conservation of momenta $p_{2}\oplus p_{3}\oplus S(p)=0$ gives
the same four possibilities. There is the additional possibility $p_{2}\oplus
S(p)\oplus p_{3}=0$ which however reduces to the previous two possibilities if
we use braiding [32]. We will assume that this is the case, and we will not
investigate this issue further in this paper.

Therefore, the equations (17), (18) are all we need to take into
consideration. We have therefore

$\displaystyle
i\Pi_{1}(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{(S(k)\oplus
p)^{2}-m^{2}+i\epsilon}$ (19) $\displaystyle
i\Pi_{2}(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{(p\oplus
S(k))^{2}-m^{2}+i\epsilon}$ (20) $\displaystyle
i\Pi_{3}(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{(k\oplus
p)^{2}-m^{2}+i\epsilon}$ (21) $\displaystyle
i\Pi_{4}(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{(p\oplus
k)^{2}-m^{2}+i\epsilon}$ (22)

We now need to expand them. We proceed as follows (we show the procedure for
the first propagator, and the others are performed in the same way)

$\displaystyle\frac{i}{(S(k)\oplus p)^{2}-m^{2}+i\epsilon}$
$\displaystyle\approx\frac{i}{(S(k)\oplus
p)^{2}-m^{2}+i\epsilon}\Big{|}_{\frac{1}{\kappa}=0}+\frac{1}{\kappa}\frac{\partial}{\partial\frac{1}{\kappa}}\frac{i}{(S(k)\oplus
p)^{2}-m^{2}+i\epsilon}\Big{|}_{\frac{1}{\kappa}=0}$
$\displaystyle+\frac{1}{2\kappa^{2}}\frac{\partial^{2}}{\partial\left(\frac{1}{\kappa}\right)^{2}}\frac{i}{(S(k)\oplus
p)^{2}-m^{2}+i\epsilon}\Big{|}_{\frac{1}{\kappa}=0}$ (23)

Taking as example the first propagator, we have therefore

$\displaystyle\frac{i}{(S(k)\oplus p)^{2}-m^{2}+i\epsilon}$
$\displaystyle=\frac{i}{(p-k)^{2}-m^{2}+i\epsilon}+\frac{1}{\kappa}\frac{i\Delta_{1}[(S(k)\oplus
p)^{2}]}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}$
$\displaystyle+\frac{1}{\kappa^{2}}\left[\frac{i(\Delta_{1}[(S(k)\oplus
p)^{2}])^{2}}{[(p-k)^{2}-m^{2}+i\epsilon]^{3}}+\frac{i\Delta_{2}[(S(k)\oplus
p)^{2}]}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}\right]$ (24)

Notice that the additional minus signs is already included in the
$\Delta_{1}$s, and the factor $1/2$ coming from the second derivative is
already included in the $\Delta_{2}$s together with the additional minus sign.
The total amplitude is therefore given by

$\displaystyle
i\Pi_{TOT}(p^{2})=\frac{i\Pi_{1}(p^{2})+i\Pi_{2}(p^{2})+i\Pi_{3}(p^{2})+i\Pi_{4}(p^{2})}{4}$
(25)

where the factor $1/4$ is needed to get the correct $\kappa\rightarrow\infty$
limit, and we get

$\displaystyle i\Pi_{TOT}(p^{2})$ $\displaystyle=i\Pi_{U}(p^{2})$ (26)
$\displaystyle+\frac{g^{2}}{\kappa}\int\frac{d^{4}k}{(2\pi)^{4}}f_{1}(k)\frac{1}{k^{2}-m^{2}+i\epsilon}\frac{1}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}$
(27)
$\displaystyle+\frac{g^{2}}{\kappa^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}f_{21}(k)\frac{1}{k^{2}-m^{2}+i\epsilon}\frac{1}{[(p-k)^{2}-m^{2}+i\epsilon]^{3}}$
(28)
$\displaystyle+\frac{g^{2}}{\kappa^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}f_{22}(k)\frac{1}{k^{2}-m^{2}+i\epsilon}\frac{1}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}},$
(29)

where

$\displaystyle f_{1}(k)=\frac{1}{4}\left[\Delta_{1}[(S(k)\oplus
p)^{2}]+\Delta_{1}[(p\oplus S(k))^{2}]+\widehat{\Delta}_{1}[(k\oplus
p)^{2}]+\widehat{\Delta}_{1}[(p\oplus k)^{2}]\right]$ (30) $\displaystyle
f_{21}(k)=\frac{1}{4}\left[(\Delta_{1}[(S(k)\oplus
p)^{2}])^{2}+(\Delta_{1}[(p\oplus
S(k))^{2}])^{2}+(\widehat{\Delta}_{1}[(k\oplus
p)^{2}])^{2}+(\widehat{\Delta}_{1}[(p\oplus k)^{2}])^{2}\right]$ (31)
$\displaystyle f_{22}(k)=\frac{1}{4}\left[\Delta_{2}[(S(k)\oplus
p)^{2}]+\Delta_{2}[(p\oplus S(k))^{2}]+\widehat{\Delta}_{2}[(k\oplus
p)^{2}]+\widehat{\Delta}_{2}[(p\oplus k)^{2}]\right].$ (32)

If we instead consider the case of one incoming particle with momentum $-p$
(to be more precise, the momentum $-p$ here needs to be interpreted as $p^{*}$
in [26]), then all the reasoning is the same, except that in eq. (17), (18)
one must switch $p\mapsto-S(p)$ (or, to use the notation in [26], we perform
the switch $p\mapsto S(p^{*})$). One therefore obtains an equivalent
expression to eq. (26), but with numerators given by

$\displaystyle g_{1}(k)$
$\displaystyle=\frac{1}{4}\big{[}\Delta_{1}[(S(k)\oplus[-S(p)])^{2}]+\Delta_{1}[([-S(p)]\oplus
S(k))^{2}]$
$\displaystyle+\widehat{\Delta}_{1}[(k\oplus[-S(p)])^{2}]+\widehat{\Delta}_{1}[([-S(p)]\oplus
k)^{2}]\big{]}$ (33) $\displaystyle g_{21}(k)$
$\displaystyle=\frac{1}{4}\big{[}(\Delta_{1}[(S(k)\oplus[-S(p)])^{2}])^{2}+(\Delta_{1}[([-S(p)]\oplus
S(k))^{2}])^{2}$
$\displaystyle+(\widehat{\Delta}_{1}[(k\oplus[-S(p)])^{2}])^{2}+(\widehat{\Delta}_{1}[([-S(p)]\oplus
k)^{2}])^{2}\big{]}$ (34) $\displaystyle g_{22}(k)$
$\displaystyle=\frac{1}{4}\big{[}\Delta_{2}[(S(k)\oplus[-S(p)])^{2}]+\Delta_{2}[([-S(p)]\oplus
S(k))^{2}]$
$\displaystyle+\widehat{\Delta}_{2}[(k\oplus[-S(p)])^{2}]+\widehat{\Delta}_{2}[([-S(p)]\oplus
k)^{2}]\big{]}.$ (35)

Despite the fact that the expansion of each of the $\Delta$ in this case is
different from the previous case with incoming momentum $S(p)$, after summing
their contributions one can show that

$\displaystyle g_{1}(k)=-f_{1}(k),\qquad g_{21}(k)=f_{21}(k),\qquad
g_{22}(k)=f_{22}(k).$ (36)

We see that in general, after expanding in powers of $1/\kappa$, we are left
with integrands whose singularity structure is similar to the undeformed case,
with the only difference being that some of the propagators have exponents
different from one. We now present an argument which shows the validity of the
Cutkosky cutting rules for each term in the expanded amplitude above, and we
then reduce the computation of the cuts for the amplitude $i\Pi_{TOT}$ in eq.
(26) to simpler amplitudes.

##  V Cutkosky rule in the deformed context

The Cutkosky rules by definition work on non-deformed QFT. Notice that in the
non-deformed context they hold even in the presence of a cut-off if the cut
propagators are still degenerate in the integration domain (and therefore if
the argument of the respective Dirac delta after the cut can still go to zero
in the integration domain). Indeed, the imaginary part does not depend on the
cut-off (recall that counterterms coming from a Hermitian Lagrangian can only
be real). Therefore, even in the presence of a cut-off, the imaginary part of
a diagram can be computed by considering all the possible cuts.

We now show that the Cutkosky rules also correctly reproduce the singularity
structure of each term in the $1/\kappa$ expansion of a $\kappa$-deformed
amplitude in a $g\phi\varphi^{2}$ interaction model. The key point of the
argument below lies in the fact that in the deformed case, one needs to
introduce a cut-off to justify in a clearer way the expansion in powers of
$1/\kappa$, so that each momentum is bounded by $\kappa$. Contrary to the non-
deformed case, however, where one is free to move the value of the cut-off
$\Lambda$ so that all the propagators can go on-shell, in the deformed context
this is not true in general, since the cut-off cannot go above $\kappa$ to
have a well-defined expansion in powers of $1/\kappa$.

Using this fact, we now consider a non-deformed model with interactions
$g\phi\varphi^{2}$ and an additional interaction $G\varphi\Phi^{2}$ where
$\Phi$ has a mass $M$ such that $M/\kappa>1$. We concentrate on 1-loop
amplitudes, but the reasoning can be generalized to more than one loop.
Consider the following diagram.

The bold lines represent propagators containing the mass $M$. Its amplitude
(with external legs excluded) reads

$\displaystyle i\Pi=(ig)^{2}(iG)^{2}\int dk\,dl\,\,N(k,l,p,m,M)$
$\displaystyle\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}\times$
$\displaystyle\times\frac{i}{l^{2}-M^{2}+i\epsilon}\frac{i}{(l-k)^{2}-M^{2}+i\epsilon}$
(37)

where all loop momenta are limited in modulus by $\kappa$, and $N$ is some
analytic function in the masses, loop momenta, and external momenta.

There are only two types of cut, represented by the vertical lines in the
figure below.

Since we are considering the case where the mass $M$ of the bold loop is above
the cut-off scale, the cutting rules still continue to hold, but this time the
cut passing through the bold loop contributes as zero. Indeed, none of its
propagators can go on-shell, so that the Dirac delta associated to the cut is
identically zero. This is reflected in the full amplitude, where the two
propagators no longer generate singularities and can be absorbed in the
analytic numerator. The imaginary part is therefore only due to the
propagators of the outer loop. Therefore, the Cutkosky cutting rules still
capture correctly the singularity structure of deformed diagrams.

The above diagram only provides justification for the first term in the
$1/\kappa$ expansion of a 1-loop deformed diagram in terms of a 2-loop non-
deformed diagram. In general, however, higher order coefficients in the
expansion of a $\kappa$-deformed amplitude in powers of $1/\kappa$ can contain
one propagator being exponentiated to some integer power $n$, and the second
one to the power $m$. In this case, one can reproduce the same reasoning as
above, relating this deformed diagram to the non-deformed one represented in
the picture below.

The top side contains $m-1$ additional heavy loops, and the lower one $n-1$
additional heavy loops. Reasoning as before, and setting $M/\kappa>1$, shows
that Cutkosky rules also reproduce the singularity structure of this diagram.
Since this holds for any $m$ and $n$, it holds at any order in perturbation in
powers of $1/\kappa$.

Although the above argument is useful to establish the validity of the cutting
rules, its usefulness is limited in actual computations. To avoid this, we now
introduce a procedure that shows in a different way the fact that the cutting
rules reproduce the singularity structure of the expanded $\kappa$-deformed
amplitude. The key point is to reduce each term of the expansion of the
deformed amplitude to simpler terms, using algebraic and analytic identities,
such that the validity of the Cutkosky rules is again manifest. Furthermore,
using these simpler terms, the actual computations in concrete cases is much
more simplified. We limit our procedure to the second order in the expansion
in $1/\kappa$, but the reasoning can be further generalized to higher orders.

##  VI Simplified Cutkosky rule in the deformed context

Consider the case of a general, non-trivial numerator $f(k,p,m)$, so that we
have

$\displaystyle i\widetilde{\Pi}_{d}(p^{2})$
$\displaystyle=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\,f(k,p,m)\frac{i}{k^{2}-m^{2}+i\epsilon}\frac{i}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}$
(38)
$\displaystyle=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\,f(k,p,m)\frac{i}{k^{2}-m^{2}+i\epsilon}\left(\frac{\partial}{\partial
m^{2}}\frac{i}{(p-k)^{2}-m^{2}+i\epsilon}\right).$ (39)

Recall that any function can be written as the sum of its even and odd part
under exchange $k\mapsto p-k$ as follows (we are still considering functions
which also depend on $p$ and $m$, but for simplicity we keep these additional
dependence implicit)

$\displaystyle
f(k)=\underbrace{\frac{f(k)+f(p-k)}{2}}_{E(k)}+\underbrace{\frac{f(k)-f(p-k)}{2}}_{O(k)}$
(40) $\displaystyle E(k)\xrightarrow{k\rightarrow p-k}E(p-k)=E(k),\qquad
O(k)\xrightarrow{k\rightarrow p-k}O(p-k)=-O(k).$ (41)

Let us also use the convention

$\displaystyle A=(p-k)^{2}-m^{2}+i\epsilon,\qquad B=k^{2}-m^{2}+i\epsilon.$
(42)

It is clear that

$\displaystyle\frac{1}{A^{2}}=\frac{\partial}{\partial
m^{2}}\frac{1}{A},\qquad\frac{1}{B^{2}}=\frac{\partial}{\partial
m^{2}}\frac{1}{B}.$ (43)

Integrating by parts one can show that

$\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{E(k)}{A^{2}B}=\frac{g^{2}}{2}\frac{\partial}{\partial
m^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial
f(k)}{\partial m^{2}}\frac{1}{AB}.$ (44)

We are only left with the need to evaluate the odd part of the integral, i.e.

$\displaystyle g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{O(k)}{A^{2}B}.$ (45)

To simplify the reasoning, we first change variables $k\mapsto\frac{p+k}{2}$.
Under this change of coordinates, the symmetry properties under $k\mapsto p-k$
send $\frac{p+k}{2}\mapsto\frac{p-k}{2}$, and therefore $k\mapsto-k$. Terms
odd (even) under exchange $\kappa\mapsto p-k$ are now odd (even) under
$k\mapsto-k$. The new integrand is therefore

$\displaystyle\frac{g^{2}}{2^{4}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\widehat{O}(k)}{\widehat{A}^{2}\widehat{B}}.$
(46)

We now notice that $\widehat{O}(k)$ is a scalar (there are no free indices).
It is also assumed to be polynomial in the loop and external momenta.
Therefore one must have $\widehat{O}(k)=(pk)\widehat{E}(k)$ where
$\widehat{E}(k)$ is a n even function under $k\mapsto-k$. We now use the
identity

$\displaystyle\frac{pk}{\widehat{A}^{2}\widehat{B}}=\frac{1}{\widehat{A}^{2}}-\frac{1}{\widehat{A}\widehat{B}}.$
(47)

One can rewrite the integral containing only $A^{2}$ in the denominator in
terms of derivative of integrals with only $A$, which cannot have any
imaginary part since they are tree level. Therefore, since our aim is to apply
the Cutkosky rule to compute the imaginary part, we can ignore those
contributions. To restore the symmetry of the form $k\mapsto p-k$, it is
sufficient to change variables once again with $k\mapsto 2k-p$. In other words
(keeping in mind that we are only interested in the imaginary part) we have

$\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B}=\frac{g^{2}}{2}\frac{\partial}{\partial
m^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{\partial
f(k)}{\partial m^{2}}+2\frac{O(k)}{2pk-p^{2}}\right)\frac{1}{AB}.$ (48)

Notice that, in light of the computations in section III, all the quantities
in the RHS are well defined.

One can proceed along the same line to obtain a similar formula for the case
in which the denominator is of the form $A^{3}B$. In fact, using the same
short-hand notation, we have

$\displaystyle
i\widetilde{\Pi}_{d}(p^{2})=g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{3}B}$
(49)

and we will indicate $\partial:=\frac{\partial}{\partial m^{2}}$ for brevity.
We have

$\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{3}B}=\frac{g^{2}}{2}\partial\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B}-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial
f(k)}{A^{2}B}-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B^{2}}.$
(50)

Thanks to eq. (48) we know how to compute the first two terms. For the last
term, one can notice that

$\displaystyle-2\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B^{2}}$
$\displaystyle=-\partial^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}+\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial^{2}f(k)}{AB}+2\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial
E(k)}{A^{2}B}+\int\frac{d^{4}k}{(2\pi)^{4}}\frac{E(k)}{A^{3}B}.$ (51)

Multiplying both sides by $1/4$ and recalling the definition of even part in
eq. (40) one can substitute the resulting expression into eq. (50). The
resulting terms can now all be evaluated, with the exception of a remaining
term with integrand $E/A^{3}B$. This can however be transferred to the LHS and
switched for an odd term using the fact that if $f=C+E/2$, then $f=C-O/2$
(where $f=E+O$ and $C$ represents other terms). Again notice that we can
repeat the reasoning of the previous case, since we can write

$\displaystyle\frac{O(k)}{A^{3}B}=\frac{O(k)}{2pk-p^{2}}\left(\frac{1}{A^{3}}-\frac{1}{A^{2}B}\right).$
(52)

Once again, we can ignore the first term because it does not have any
imaginary part, since it can be written in terms of derivatives of tree
diagrams. Therefore, we have the final formula (which we recall is valid for
the computation of the imaginary part)

$\displaystyle g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{3}B}$
$\displaystyle=g^{2}\partial\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B}-\frac{g^{2}}{2}\partial^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}+\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial^{2}f(k)}{AB}$
$\displaystyle+g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(-\partial
f(k)+2\partial E(k)+\frac{O(k)}{2pk-p^{2}}\right)\frac{1}{A^{2}B}.$ (53)

Notice that in each of the terms in the RHS, the singularity structure is
simpler than the one in the RHS, and in particular is exactly the same
singularity structure of a 1-loop non-deformed correction to the propagator
(with a different numerator, which however does not influence the
singulariries). The validity of the Cutkosky rules is therefore once again
manifest. We now first show an example of an explicit check of the validity of
eq. (48) (which highlights how much simpler are the computations using the
cutting rules), and we then use both identities above to compute the full
imaginary part of the $\kappa$-deformed 1-loop correction up to second order
in $1/\kappa$.

###  VI.1 Example of checking the deformed Cutkosky rules

We can show here a quick check of eq. (48) in the simple case where the
numerator is given by $k^{2}$. We start from

$\displaystyle
i\Pi_{d}(p^{2})=(ig)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}k^{2}\underbrace{\frac{i}{k^{2}-m^{2}+i\epsilon}}_{B}\underbrace{\frac{i}{[(p-k)^{2}-m^{2}+i\epsilon]^{2}}}_{A}.$
(54)

This time the numerator has an even and an odd part under the exchange
$k\mapsto p-k$

$\displaystyle E(k)=\frac{k^{2}+(p-k)^{2}}{2},\qquad
O(k)=\frac{k^{2}-(p-k)^{2}}{2}=\frac{1}{2}(2pk-p^{2}),$ (55)

so that the above formula in eq. (48) simplifies to

$\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{A^{2}B}=\frac{g^{2}}{2}\frac{\partial}{\partial
m^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{AB}-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{AB}.$
(56)

The second term is exactly as the non-deformed case, with an additional
$-\frac{1}{2}$, so that using eq. (15) we get

$\displaystyle\Im\left(-\frac{g^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{AB}\right)=-\frac{1}{2}\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}.$
(57)

For the first term, if we ignore the derivative we see that this is just the
same as eq. (11) with an additional $m^{2}/2$ in front, so that

$\displaystyle\Im[iI]=\frac{m^{2}}{2}\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}.$
(58)

Taking the first derivative of eq. (58) with respect to $m^{2}$ and adding it
to eq. (57) we get

$\displaystyle\Im(i\Pi_{d}(p^{2}))=-\frac{g^{2}}{16\pi}\frac{m^{2}}{p^{2}}\left(\sqrt{1-4\frac{m^{2}}{p^{2}}}\right)^{-1}$
(59)

because the other two contributions cancel each other out.

To compute the imaginary part directly, we need to merge the denominators
using the Feynman trick, and then perform both the shift $k\mapsto k+p(1-x)$
and perform the Wick rotation. After these canonical steps we are left with

$\displaystyle
i\Pi(p^{2})=-ig^{2}\int_{0}^{1}dx\,\,2(1-x)\int\frac{d^{4}k}{(2\pi)^{4}}\frac{-k_{E}^{2}+p^{2}(1-x)^{2}}{(k_{E}^{2}+\Delta)^{3}}.$
(60)

We can compute the two integrals separately, starting with the one with
$k_{E}^{2}$ at the numerator. We call this integral $I_{0}$. The momentum
space integral can be computed with eq. (5) with $a=1$ and $b=3$. Therefore
the momentum space integral is again given by eq. (5), so we can follow the
same reasoning, and we end up with

$\displaystyle
i\Pi(p^{2})=-i\frac{g^{2}}{(4\pi)^{2}}\int_{0}^{1}dx\,\,2(1-x)\log(m^{2}-p^{2}x(1-x)).$
(61)

By changing $x\mapsto 1-x$ one can show that

$\displaystyle-i\frac{g^{2}}{(4\pi)^{2}}\int_{0}^{1}dx\,\,2x\log(m^{2}-p^{2}x(1-x))=-i\frac{g^{2}}{(4\pi)^{2}}\int_{0}^{1}dx\log(m^{2}-p^{2}x(1-x))$
(62)

and this is the same integral as the one encountered in section III. Therefore
the imaginary part of the term with $k_{E}^{2}$ in the numerator is

$\displaystyle\Im(I_{0})=+\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}.$
(63)

The term with $p^{2}$ at the numerator instead is given by $a=0$ and $b=3$ in
eq. (5) so that we need to compute the imaginary part of

$\displaystyle-i\frac{g^{2}}{32\pi^{2}}\int_{0}^{1}dx\,\,2x^{3}\frac{p^{2}}{m^{2}-p^{2}x(1-x)}$
(64)

where we switched $x\mapsto 1-x$ to simplify the expression. Using the
notation

$\displaystyle
I_{1}:=-i\frac{g^{2}}{32\pi^{2}}\int_{0}^{1}dx\,\,2x^{3}\frac{p^{2}}{m^{2}-p^{2}x(1-x)},\qquad
I_{2}:=-i\frac{g^{2}}{32\pi^{2}}\int_{0}^{1}dx\,\,\frac{2x}{m^{2}-p^{2}x(1-x)},$
(65) $\displaystyle
I_{3}:=-i\frac{g^{2}}{32\pi^{2}}\int_{0}^{1}dx2\log(m^{2}-p^{2}x(1-x))+4-2\log(m^{2}),$
(66)

one can show that

$\displaystyle\Im(I_{1})=-\Im(I_{3})+m^{2}\Im(I_{2})$ (67)

where

$\displaystyle\Im(I_{3})=+\frac{g^{2}}{16\pi}\sqrt{1-4\frac{m^{2}}{p^{2}}}\qquad\Im(I_{2})=-\frac{g^{2}}{16\pi}\frac{1}{p^{2}}\left(\sqrt{1-4\frac{m^{2}}{p^{2}}}\right)^{-1}.$
(68)

Adding all the contributions results in

$\displaystyle\Im(I_{0})+\Im(I_{1})=-\frac{g^{2}}{16\pi}\frac{m^{2}}{p^{2}}\left(\sqrt{1-4\frac{m^{2}}{p^{2}}}\right)^{-1}$
(69)

which is exactly the same result obtained in eq. (59) with the Cutkosky rule,
as expected.

##  VII $\kappa$-deformed imaginary part of the 1-loop correction

We now apply our relations (48) and (VI) to compute the full imaginary part of
the 1-loop correction to the propagator, up to second order in $1/\kappa$. In
the non-deformed case, one could perform the computations in the centre of
mass (c.o.m.) frame because of the trivial numerator. In the deformed case, we
will see that the numerator (after the expansion in powers of $1/\kappa$) is
not a Lorentz scalar in general, and therefore one should perform the
computations in an arbitrary frame in order to get the most general result.

Here, however, we will perform the full computations in the c.o.m.. Doing so,
we will see that the deformed contributions is proportional to
$p^{2}/\kappa^{2}$. We will then argue that the full result in an arbitrary
frame must be of the same type, containing an additional term proportional
$p_{0}^{2}/\kappa^{2}$.

In order to get the full amplitude, we will take the average of the two
amplitudes where the incoming momentum is $S(p)$ and $-p$. In fact, from a
phenomenological point of view, we can only see a particle moving towards us
at a certain speed, and we do not know whether its momentum is of the kind
$-p$ or $S(p)$.

###  VII.1 Centre of mass frame

In the c.o.m. frame we have

$\displaystyle f_{1}(k)=0,$ (70) $\displaystyle
f_{21}(k)=2\left(k_{1}^{2}p_{0}+k_{2}^{2}p_{0}+k_{3}^{2}p_{0}\right)^{2},$
(71) $\displaystyle
f_{22}(k)=\frac{1}{4}\left(2k_{0}^{3}p_{0}+k_{0}^{2}\left(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-4p_{0}^{2}\right)+2k_{0}p_{0}^{3}+k_{1}^{2}p_{0}^{2}+k_{2}^{2}p_{0}^{2}+k_{3}^{2}p_{0}^{2}\right),$
(72)

and therefore we only have the $1/\kappa^{2}$ contribution to the amplitude.
Since there is no explicit dependence on the mass $m$, we can use the
simplified relations

$\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B}=\frac{g^{2}}{2}\frac{\partial}{\partial
m^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}-g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{O(k)}{2pk-p^{2}}\right)\frac{1}{AB},$
(73) $\displaystyle
g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{3}B}=g^{2}\partial\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{A^{2}B}-\frac{g^{2}}{2}\partial^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f(k)}{AB}+g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{O(k)}{2pk-p^{2}}\right)\frac{1}{A^{2}B}.$
(74)

We start with the computations of eq. (73), since we will need it for the
computation of eq. (74). The starting point is the formula

$\displaystyle 2\Im[iA]=\frac{g^{2}}{16\pi^{2}}\int
d^{3}k\frac{N(k_{0},k_{0}+q_{0},\mathbf{k},0)}{k_{0}q_{0}}\delta(q_{0}+k_{0}-p_{0}).$
(75)

where $N$ stands for a generic numerator with $\mathbf{p}=0$, and proceeding
as before in section III we have $p_{\mu}=(p_{0},0,0,0)$,
$k_{\mu}=(\sqrt{|\mathbf{k}|^{2}+m^{2}},\mathbf{k})$, and
$q_{\mu}=(\sqrt{|\mathbf{k}|^{2}+m^{2}},-\mathbf{k})$, so that the above
integral becomes (we also pass to polar coordinates in momentum space)

$\displaystyle 2\Im[iA]$ $\displaystyle=\frac{g^{2}}{32\pi^{2}}\int
d|\mathbf{k}|\,\sin\theta
d\theta\,d\phi\,N(\sqrt{|\mathbf{k}|^{2}+m^{2}},2\sqrt{|\mathbf{k}|^{2}+m^{2}},|\mathbf{k}|,\theta,\phi,0)\times$
$\displaystyle\times\frac{|\mathbf{k}|}{\sqrt{|\mathbf{k}|^{2}+m^{2}}}\delta\left(|\mathbf{k}|-\sqrt{\frac{p_{0}^{2}}{4}-m^{2}}\right).$
(76)

Using this relation, we can compute the imaginary part of the terms in eq.
(28) and (29) in the amplitude.

Starting from eq. (28), one can show that the first two terms in the RHS of
eq. (74) are equal and opposite, while the third is zero because $O_{21}=0$
identically in the c.o.m. frame. Therefore, we get that the contribution of
eq. (28) to the amplitude vanishes

$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f_{21}(k)}{A^{3}B}=0.$ (77)

We can therefore compute the contribution given by eq. (29). One can show that
the contributions to the RHS of eq. (73) are given by

$\displaystyle\Im\left(\frac{g^{2}}{2}\frac{\partial}{\partial
m^{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{f_{22}(k)}{AB}\right)=\frac{g^{2}\left(60m^{2}p_{0}-19p_{0}^{3}\right)}{1024\pi\sqrt{p_{0}^{2}-4m^{2}}},$
(78)
$\displaystyle\Im\left(-g^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{O(k)}{2pk-p^{2}}\right)\frac{1}{AB}\right)=-\frac{g^{2}\left(p_{0}^{2}-4m^{2}\right)^{3/2}}{512\pi
p_{0}}.$ (79)

Summing the results of eq. (78) and (79) we have the full result of the
deformed contribution to the imaginary part of the 1-loop correction to the
scalar propagator. One can express the full deformed imaginary part
$\Im[i\Pi_{TOT}(p^{2})]$ in terms of the non deformed one
$\Im[i\Pi_{U}(p^{2})]$ (whose expression can be found in eq. (15)) as follows

$\displaystyle\Im[i\Pi_{TOT}(p^{2})]=\Im[i\Pi_{U}(p^{2})]\left(1+\frac{1}{\kappa^{2}}\frac{32m^{4}-76m^{2}p^{2}+21p^{4}}{256m^{2}-64p^{2}}\right).$
(80)

Notice that we have substituted $p_{0}$ with $p$ since in the c.o.m. they are
the same. This expression can be further simplified with some algebra,
arriving at the final result

$\displaystyle\Im[i\Pi_{TOT}(p^{2})]=\Im[i\Pi_{U}(p^{2})]\left[1-\frac{p^{2}}{\kappa^{2}}\left(\frac{512\pi^{2}}{g^{4}}(\Im[i\Pi_{U}(p^{2})])^{2}+\frac{g^{4}}{64\pi^{2}(\Im[i\Pi_{U}(p^{2})])^{2}}+\frac{1}{8}\right)\right].$
(81)

In light of eq. (36), the same result holds for an incoming particle of
momentum $-p$, and therefore the above result is also true for the average of
the two cases.

###  VII.2 General frame and decay widths

Notice that we do not get lower order corrections moving away from the c.o.m.
frame, since because of eq. (36) we have that the $1/\kappa$ contribution
identically vanishes in any frame when taking the average. The only difference
with the c.o.m. frame is the presence of additional terms in the numerator
proportional to $p_{0}$ and $\mathbf{p}=\sqrt{p^{2}-p_{0}^{2}}$. Therefore,
for dimensional reasons, the general correction can only have the form

$\displaystyle\Im[i\Pi_{TOT}(p^{2})]=\Im[i\Pi_{U}(p^{2})]\left[1+\alpha\frac{p^{2}}{\kappa^{2}}+\beta\frac{p_{0}^{2}}{\kappa^{2}}\right],$
(82)

where $\alpha$ and $\beta$ are coefficients whose computation is beside the
scope of the present work, although (ignoring the boost factors $\gamma$) we
expect them to be of the same order of magnitude as the coefficients in the
c.o.m. frame. To understand in detail their dependence on the boost factor
$\gamma$ would require their full computation. For single diagrams, the
$\alpha$ and $\beta$ coefficients will be in general different, but as we saw
above, if we sum all the contributions then $\alpha$ and $\beta$ are the same,
regardless of whether the incoming particle has momentum $S(p)$ or $-p$.

This allows us to put some first stringent limits on the behaviour of the
decay width of unstable particles in the model described in [20, 26, 27].
Indeed, by using the Dyson resummation for the above 1-loop correction, we get

$\displaystyle
i\Delta=\frac{1}{p^{2}-m^{2}}\sum_{i=0}^{\infty}\left(-\frac{\Pi_{TOT}(p^{2})}{p^{2}-m^{2}}\right)=\frac{1}{p^{2}-m^{2}+\Pi_{TOT}(p^{2})},$
(83)

where the quantity $\Pi_{TOT}(p^{2})$ will contain both a real and an
imaginary part. The real part is finite, since the zeroth order in $1/\kappa$
is the finite non-deformed term, and each subsequent integral in perturbation
series has a cut-off at $\kappa$. The imaginary part is the one we computed
above. The relation

$\displaystyle m\Gamma=\Im\left(\Pi_{TOT}(m^{2})\right)$ (84)

relates the decay width to the imaginary part of the 1-loop correction. One
immediately sees, therefore, that the 1-loop contributions to the decay width
are at best of the order of $p_{0}^{2}/\kappa^{2}$, and therefore negligible.

##  VIII Conclusions and discussion

In this paper we showed the validity of the Cutkosky cutting rules order by
order in the expansion of a deformed amplitude (for definiteness, and since
our discussion is fairly general, we call the deformation parameter
$\lambda$). In general, after expansion in $\lambda$, these amplitudes will
have a denominator with propagators where canonical conservation of momentum
is applied, but with either (or both) propagators exponentiated to a power
different than one. We showed that by truncating the integration domain (a
requirement needed to have a well-defined expansion in $\lambda$), the
Cutkosky rules still reproduce correctly the singularity structure of the
terms in the expansion of the deformed amplitude. The argument is based on
relating each term in the expansion in $\lambda$ to non-deformed amplitude
containing a particle of high mass. One could in principle apply directly the
cutting rules and proceed with the computations. It is instead much simpler to
first reduce the powers of the propagator in the denominator by using several
algebraic and analytic identities. Although we showed how to do this directly
only up to second order in $\lambda$, the reasoning can be easily generalized,
and it provides a different way to show that indeed the cutting rules still
capture the singularity structure of the expansion in $\lambda$ of the
deformed amplitude. Pragmatically, using this approach one reduces the
computation of the imaginary part to simpler amplitudes. Iterating the
procedure in the same way until there are only terms with just canonical
propagators, the application of the Cutkosky rules become trivial.

Notice that the use of the cutting rules simplifies dramatically the
computation of the imaginary part of loop corrections to the propagator. If
one were to proceed with the canonical computations, one would first need to
merge the denominator using the Feynman trick described by the relation

$\displaystyle\frac{1}{A^{a}B^{b}}=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{1}dx\,\,\frac{x^{a-1}(1-x)^{b-1}}{[Ax+B(1-x)]^{a+b}}.$
(85)

After the shift in the loop momentum, one would first need to compute the
momentum-space integrals, and then integrate in $x$. After this, one needs to
extract the imaginary part from the result. Due to the fact that numerators
become quickly very long after expansion in $\lambda$, this makes it very
difficult to extract the imaginary part of even reasonably simple corrections,
as was already noted in [30]. The approach presented in this paper makes for
an easier computation of higher-order terms in the $\lambda$ expansion.

Due to this simplicity, we were able to show an example of calculation of the
imaginary part of a diagram in a particular model. We chose this model also
because, in previous publications [27], it was qualitatively argued that the
first-order correction due to deformation of the imaginary part of the 1-loop
diagram should be negligible. This was already partially checked in [30]
where, for simplicity, only the imaginary part coming from the logarithms was
studied. Here we confirmed in full generality that this is still the case when
considering the full amplitude. This is an important result from a
phenomenological point of view. Indeed, the fact that the decay width
$\Gamma=1/\tau$ (where $\tau$ is the decay time of the unstable particle) is
not affected by deformation up to negligible terms proportional to
$1/\kappa^{2}$, makes it evident that the deformation effects described in
[27, 24] are purely due to the behaviour under $CPT$ transformation of the
fields.

From both an experimental and phenomenological point of view, the next natural
step would be to extend the above computations to more loops. In this context,
however, the non-planar graphs significantly complicate the matter.
Furthermore, it would be interesting to consider in detail the implication of
the above arguments about unitarity for general deformed amplitudes. We will
address these interesting points in future publications.

## Acknowledgements

I would like to thank Jerzy Kowalski-Glikman and Wojciech Wiślicki for their
guidance and their help, as well as for reading through the preliminary
versions of this manuscript, providing helpful advice. I would also like to
thank Alice Boldrin, for both her patience and her help in the writing of this
paper. This work was supported by funds provided by the National Science
Center, project no. 2019/33/B/ST2/00050.

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# Continuous Transitions Between Quantum and Classical Electrodynamics

Partha<EMAIL_ADDRESS>
The National Academy of Sciences, India,
5 Lajpatrai Road, Allahabad 211002, India

###### Abstract

The Maxwell equations in the presence of sources are first derived without
making use of the potentials and the Hamilton-Jacobi equation for classical
electrodynamics is written down. The manifestly gauge invariant theory is then
quantized to write down the Hamilton-Jacobi equation in quantum
electrodynamics. Finally, an interpolating field theory is proposed that
describes continuous transitions between quantum and classical
electrodynamics. It is shown that energy flow lines are identical for quantum
and classical fields in the case of the double-slit arrangement but differ in
the case of vortex beams.

##  1 Classical Electrodynamics

###  1.1 Maxwell Equations in Hamiltonian Form

Consider the conventional free Hamiltonian in classical electrodynamics

$H_{0}[E,B]=\int d^{3}x^{\prime}\rho_{0}^{\prime}=\int
d^{3}x^{\prime}\frac{1}{2}\left[\epsilon_{0}\vec{E}^{\prime}.\vec{E}^{\prime}+\mu_{0}^{-1}\vec{B}^{\prime}.\vec{B}^{\prime}\right]$
(1)

where $E^{\prime}_{i}\equiv E_{i}(\vec{x}^{\prime},t^{\prime})$ and
$B^{\prime}_{i}\equiv B_{i}(\vec{x}^{\prime},t^{\prime})$ are the electric and
magnetic fields. To obtain the full set of Maxwell equations in the presence
of sources, consider the change in the Hamiltonian

$\displaystyle H[E,B,t]$ $\displaystyle=$ $\displaystyle
H_{0}[E,B]+\int_{0}^{t}\vec{E}.\vec{j}_{q}d^{3}xdt=\int
d^{3}x^{\prime}\rho^{\prime}=\int
d^{3}x^{\prime}\int_{0}^{t}\frac{\partial\rho^{\prime}}{\partial
t^{\prime}}dt^{\prime}$ (2) $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}\int_{0}^{t}\left[\mu_{0}^{-1}\vec{B}^{\prime}.\frac{\partial\vec{B}^{\prime}}{\partial
t^{\prime}}+\epsilon_{0}\vec{E}^{\prime}.\left(\frac{\partial\vec{E}^{\prime}}{\partial
t^{\prime}}+c^{2}\mu_{0}\vec{j}_{q}^{\prime}\right)\right]dt^{\prime}$ (3)

where $\vec{E}.\vec{j}_{q}dt$ is the energy density absorbed by a charge $q$
in time $dt$ due to the Lorentz force acting on it. Now define the Poynting
vector $\vec{S}=\mu_{0}^{-1}(\vec{E}\times\vec{B})$ and the functional

$\displaystyle\frac{\partial S[E,B,t]}{\partial t}$ $\displaystyle=$
$\displaystyle\int
d^{3}x^{\prime}\int_{0}^{t}dt^{\prime}\vec{\nabla}^{\prime}.\vec{S}^{\prime}$
(4) $\displaystyle=$ $\displaystyle\mu_{0}^{-1}\int
d^{3}x^{\prime}\int_{0}^{t}dt^{\prime}\left[\vec{B}^{\prime}.(\vec{\nabla}^{\prime}\times\vec{E}^{\prime})-\vec{E}^{\prime}.(\vec{\nabla}^{\prime}\times\vec{B}^{\prime})\right]$
(5)

where use has been made of the relation $\epsilon_{0}\mu_{0}=c^{-2}$ and the
vector identity

$\vec{\nabla}.(\vec{a}\times\vec{b})=\vec{b}.(\vec{\nabla}\times\vec{a})-\vec{a}.(\vec{\nabla}\times\vec{b}).$
(6)

Let us now define a new functional

$K=H[E,B,t]+\frac{\partial S}{\partial t}=0.$ (7)

If this is to hold for all functions $(E,B)$, the Maxwell equations

$\displaystyle\vec{\nabla}\times\vec{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\vec{B}}{\partial t},$ (8)
$\displaystyle\vec{\nabla}\times\vec{B}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}+\mu_{0}\vec{j}_{q},$ (9)

as well as the equation

$\frac{\partial\rho}{\partial t}+\vec{\nabla}.\vec{S}=0,$ (10)

or

$\frac{\partial\rho_{0}}{\partial
t}+\vec{\nabla}.\vec{S}=-\vec{E}.\vec{j}_{q}$ (11)

must be satisfied. The last equation follows from eqns.(2) and (4) and will be
recognized as Poynting’s theorem.

This completes the derivation of the full set of Maxwell equations in the
presence of sources without the need to introduce the vector potential
$A_{\mu}$. The procedure is manifestly gauge invariant.

Now consider the free-field Maxwell equations

$\displaystyle\vec{\nabla}\times\vec{B}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial t},$ (12)
$\displaystyle\vec{\nabla}\times\vec{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\vec{B}}{\partial t}.$ (13)

Notice that these equations can be written in the forms

$\displaystyle\dot{E}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{0}}\epsilon_{ijk}\partial_{j}\frac{\delta
H_{0}[E,B]}{\delta B_{k}}=c^{2}\epsilon_{ijk}\partial_{j}B_{k},$ (14)
$\displaystyle\dot{B}_{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{\epsilon_{0}}\epsilon_{ijk}\partial_{j}\frac{\delta
H_{0}[E,B]}{\delta E_{k}}=-\epsilon_{ijk}\partial_{j}E_{k}.$ (15)

These are therefore generalizations of Hamilton’s equations in classical
mechanics appropriate for vector field theories like Maxwell’s
Electrodynamics. They can also be written in terms of Poisson brackets if the
latter are generalized as follows:

$\displaystyle\\{E_{i},H[E,B]\\}_{PB}$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon_{0}}\epsilon_{ijk}\partial_{j}\frac{\delta
H[E,B]}{\delta B_{k}}=\dot{E}_{i},$ (16)
$\displaystyle\\{B_{i},H[E,B]\\}_{PB}$ $\displaystyle=$
$\displaystyle-\frac{1}{\epsilon_{0}}\epsilon_{ijk}\partial_{j}\frac{\delta
H[E,B]}{\delta E_{k}}=\dot{B}_{i}.$ (17)

With these generalizations $\vec{E}$ and $\vec{B}$ can be regarded as
canonically conjugate variables.

###  1.2 Hamilton-Jacobi Equation in Classical Electrodynamics

For free fields $\vec{E}=\vec{E}_{0}e^{i\omega t}$, eqn.(7) becomes

$H_{0}[E,B]+\frac{\partial S}{\partial t}=0$ (18)

with $H_{0}[E,B]$ given by (1). Setting

$E_{i}=\frac{1}{\omega\epsilon_{0}}\epsilon_{ijk}\partial_{j}\left(\frac{\delta
S}{\delta B_{k}}\right)\equiv\frac{1}{\omega\epsilon_{0}}\left(\frac{\Delta
S}{\Delta B}\right)_{i},$ (19)

in equation (18), one obtains

$\displaystyle\frac{\partial S}{\partial t}+\frac{1}{2}\int
d^{3}x^{\prime}\left[\frac{1}{\omega^{2}\epsilon_{0}}\left(\frac{\Delta
S}{\Delta
B^{\prime}}\right)^{2}+\mu_{0}^{-1}\vec{B}^{\prime}.\vec{B}^{\prime}\right]=0,$
(20) $\displaystyle\left(\frac{\Delta S}{\Delta B^{\prime}}\right)^{2}$
$\displaystyle\equiv$ $\displaystyle\left[\partial^{\prime
2}\left(\frac{\delta S}{\delta
B_{k}^{\prime}}\right)^{2}-\partial^{\prime}_{j}\partial^{\prime}_{k}\left(\frac{\delta
S}{\delta B^{\prime}_{j}}\right)\left(\frac{\delta S}{\delta
B^{\prime}_{k}}\right)\right].$ (21)

This is the Hamilton-Jacobi equation in classical electrodynamics.

##  2 Hamilton-Jacobi Equation in Quantum Electrodynamics

Choosing the operators $\hat{B}_{i}=B_{i}$ and

$\hat{E}_{i}(\vec{x},t)=\frac{i\hbar}{\omega\epsilon_{0}}\epsilon_{ilm}\partial_{l}\frac{\delta}{\delta
B_{m}(\vec{x},t)},$ (22)

one has the equal-time commutation relations

$\displaystyle\left[\hat{E}_{i}(\vec{x},t),\hat{B}_{j}(\vec{y},t)\right]$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{\epsilon_{0}}\epsilon_{ijk}\partial_{k}\delta^{3}(\vec{x}-\vec{y}),$
(23) $\displaystyle\left[\hat{E}_{i}(\vec{x},t),\hat{E}_{j}(\vec{y},t)\right]$
$\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\left[\hat{B}_{i}(\vec{x},t),\hat{B}_{j}(\vec{y},t)\right]$
$\displaystyle=$ $\displaystyle 0.$ (25)

This is a manifestly gauge invariant form of field quantization.

Using (22), the Schrödinger equation for a field functional $\Psi[B_{i},t]$
can be written as

$\displaystyle i\hbar\frac{\partial\Psi[B_{i},t]}{\partial t}$
$\displaystyle=$ $\displaystyle H\Psi[B_{i},t]=\frac{1}{2}\int
d^{3}x^{\prime}\left(-\frac{\hbar^{2}}{\omega^{2}\epsilon_{0}}\frac{\Delta^{2}}{\Delta
B^{\prime
2}}+\mu_{0}^{-1}\vec{B}^{\prime}.\vec{B}^{\prime}\right)\Psi[B_{i},t],$ (26)
$\displaystyle\frac{\Delta^{2}}{\Delta B^{\prime 2}}$ $\displaystyle=$
$\displaystyle\left[\partial^{\prime 2}\left(\frac{\delta}{\delta
B_{k}^{\prime}}\right)\left(\frac{\delta}{\delta
B_{k}^{\prime}}\right)-\partial^{\prime}_{j}\partial^{\prime}_{k}\left(\frac{\delta}{\delta
B^{\prime}_{j}}\right)\left(\frac{\delta}{\delta
B^{\prime}_{k}}\right)\right].$ (27)

Writing $\Psi$ in the polar form

$\displaystyle\Psi[B_{i},t]$ $\displaystyle=$ $\displaystyle R[B_{i},t]\,{\rm
exp}\left(\frac{i}{\hbar}S[B_{i},t]\right)$ (28)

where $R[B_{i},t]$ and $S[B_{i},t]$ are two real functionals, separating the
real and imaginary parts, and defining

$Q^{\prime}=-\frac{\hbar^{2}}{\omega^{2}\epsilon_{0}}\frac{\Delta^{2}R/\Delta
B^{\prime 2}}{R},$ (29)

one obtains the two equations

$\displaystyle\frac{\partial S}{\partial t}+\frac{1}{2}\int
d^{3}x^{\prime}\left[\frac{1}{\omega^{2}\epsilon_{0}}\left(\frac{\Delta
S}{\Delta
B^{\prime}}\right)^{2}+\mu_{0}^{-1}\vec{B}^{\prime}.\vec{B}^{\prime}+Q^{\prime}\right]$
$\displaystyle=$ $\displaystyle 0,$ (30) $\displaystyle\frac{\partial
R^{2}}{\partial t}+\frac{1}{\omega^{2}\epsilon_{0}}\int
d^{3}x^{\prime}\frac{\Delta}{\Delta B^{\prime}_{i}}\left(R^{2}\frac{\Delta
S}{\Delta B^{\prime}_{i}}\right)$ $\displaystyle=$ $\displaystyle 0.$ (31)

The first equation is the Hamilton-Jacobi equation in quantum electrodynamics,
and the second one is a functional continuity equation for $R^{2}$.

Comparing eqn.(30) with (20), we see that it differs from (20), the classical
Hamilton-Jacobi equation, only by the term $Q^{\prime}$. A great advantage of
this is therefore that all quantum effects arising from the operator nature of
the field $\hat{E}_{i}$ are neatly isolated in the quantum potential $Q$
alone.

Defining the momentum $\Pi_{i}=\frac{1}{\omega^{2}\epsilon_{0}}\Delta S/\Delta
B_{i}\equiv E_{i}$, one can rewrite eqn. (30) in the form

$\displaystyle\frac{\partial S}{\partial t}+\frac{1}{2}\int
d^{3}x^{\prime}\left[\epsilon_{0}\vec{\Pi}.\vec{\Pi}+\mu_{0}^{-1}\vec{B}.\vec{B}+Q^{\prime}\right]$
(32) $\displaystyle\equiv$ $\displaystyle\frac{\partial S}{\partial
t}+\frac{1}{2}\int
d^{3}x^{\prime}\left[\epsilon_{0}\vec{E}.\vec{E}+\mu_{0}^{-1}\vec{B}.\vec{B}+Q^{\prime}\right]$
$\displaystyle=$ $\displaystyle\frac{\partial S}{\partial t}+\int
d^{3}x^{\prime}\rho_{Q}^{\prime}=\frac{\partial S}{\partial t}+\int
d^{3}x^{\prime}\left(\rho_{cl}^{\prime}+Q^{\prime}\right)=0.$

Following the standard procedure, one can therefore show that the Poynting
vector $\vec{S}$ satisfies the equation

$\displaystyle\frac{\partial\rho_{Q}}{\partial t}+\vec{\nabla}.\vec{S}=0,$
(33)

or

$\displaystyle\frac{\partial{\rho}_{cl}}{\partial
t}+\vec{\nabla}.\vec{S}=-\frac{\partial Q}{\partial t},$ (34)

which is the Poynting theorem in quantum electrodynamics.

##  3 An Interpolating Field

Let us now postulate an interpolating nonlinear Schrödinger equation

$i\hbar\frac{\partial\Psi[B_{i},t]}{\partial t}=\frac{1}{2}\int
d^{3}x^{\prime}\left(-\frac{\hbar^{2}}{\omega^{2}\epsilon_{0}}\frac{\Delta^{2}}{\Delta
B^{\prime
2}}+\mu_{0}^{-1}\vec{B}^{\prime}.\vec{B}^{\prime}-\lambda(\theta)Q^{\prime}\right)\Psi[B_{i},t]$
(35)

where $\lambda(\theta)$ is a coupling function that is assumed to be a
continuous function of some parameter $\theta$, satisfying the conditions
$0<\lambda(\theta)\leq 1$. Notice that $\lambda(\theta)\rightarrow 0$ is the
quantum limit (eqn. (30)) and $\lambda(\theta)\rightarrow 1$ is the classical
limit in which the quantum potential density $Q$ does not contribute. Then the
modified functional Hamilton-Jacobi equation for a ‘mesostate’ of the
electromagnetic field is

$\displaystyle\frac{\partial S}{\partial t}+\int
d^{3}x^{\prime}\rho^{\prime}_{mes}$ $\displaystyle=$ $\displaystyle 0,$ (36)

with

$\displaystyle\rho_{mes}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\epsilon_{0}\vec{E}.\vec{E}+\mu_{0}^{-1}\vec{B}.\vec{B}+(1-\lambda(\theta))Q\right]$
(37) $\displaystyle=$ $\displaystyle\rho_{cl}+(1-\lambda(\theta))Q.$ (38)

Thus, equation (35) provides a basis for studying continuous transitions
between quantum and classical electrodynamics.

The Poynting theorem now takes the form

$\displaystyle\frac{\partial\rho_{mes}}{\partial t}+\vec{\nabla}.\vec{S}=0.$
(39)

Using the relation

$\vec{S}=\rho_{mes}\vec{c},$ (40)

we get

$\vec{c}=c\frac{d\vec{r}}{ds}=\frac{\vec{S}}{\rho_{mes}}$ (41)

where $ds$ is the infinitesimal arc length corresponding to an infinitesimal
path $d\vec{r}$ of energy flow. This differential equation determines the
energy-flow pattern in the field, and as is evident, this pattern changes with
the parameter $\lambda(\theta)$, giving the classical pattern in the limit
$\lambda(\theta)\rightarrow 1$ and the fully quantized pattern in the limit
$\lambda(\theta)\rightarrow 0$.

This is thus a generalization of the theory of continuous transitions between
quantum and classical mechanics formulated earlier [1, 2, 3] to the case of a
relativistic field theory.

##  4 Applications

###  4.1 The Double-slit

The energy flow lines for plane ploarized light behind a two-dimensional
Ronchi grating have been calculated in Ref. [4]. Consider a monochromatic
solution of the Maxwell equations with frquency $\omega$ given by

$\displaystyle\vec{E}(\vec{r},t)$ $\displaystyle=$
$\displaystyle\vec{E}(\vec{r})e^{-i\omega t},$ (42)
$\displaystyle\vec{H}(\vec{r},t)$ $\displaystyle=$
$\displaystyle\vec{H}(\vec{r})e^{-i\omega t}.$ (43)

It is well known that the time independent parts of the fields satisfy the
Helmholtz equations

$\displaystyle\nabla^{2}\vec{E}(\vec{r})+k^{2}\vec{E}(\vec{r})$
$\displaystyle=$ $\displaystyle 0,$ (44)
$\displaystyle\nabla^{2}\vec{H}(\vec{r})+k^{2}\vec{H}(\vec{r})$
$\displaystyle=$ $\displaystyle 0,$ (45)

where $k=\omega/c$.

Now consider a linearly polarized plane light wave propagating in the $y$
direction and incident on a grating in the $xz$ plane at $y=0$. Then, as shown
in Born and Wolf [5] and also Ref. [4], the fields diffracted by the grating
are given by

$\displaystyle\vec{H}$ $\displaystyle=$ $\displaystyle\mp
ik^{-1}Be\left(\frac{\partial\psi}{\partial
y}\hat{e}_{x}-\frac{\partial\psi}{\partial
x}\hat{e}_{y}\right)+A\psi\hat{e}_{z},$ (46) $\displaystyle\vec{E}$
$\displaystyle=$
$\displaystyle\frac{iA}{\epsilon_{0}\omega}\left(\frac{\partial\psi}{\partial
y}\hat{e}_{x}-\frac{\partial\psi}{\partial
x}\hat{e}_{y}\right)\pm\frac{kB}{\epsilon_{0}\omega}\psi\hat{e}_{z},$ (47)

where $A$ and $B$ are real constants and $\psi$ is a solution of the Helmholtz
equation

$(\nabla^{2}+k^{2})\psi(x,y)=0,$ (48)

satisfying the grating boundary conditition. The components of the time
averaged Poynting vector are given by

$\displaystyle S_{x}$ $\displaystyle=$
$\displaystyle\frac{i}{4\epsilon_{0}\omega}(A^{2}+B^{2})\left(\psi\frac{\partial\psi^{*}}{\partial
x}-\psi^{*}\frac{\partial\psi}{\partial x}\right),$ (49) $\displaystyle S_{y}$
$\displaystyle=$
$\displaystyle\frac{i}{4\epsilon_{0}\omega}(A^{2}+B^{2})\left(\psi\frac{\partial\psi^{*}}{\partial
y}-\psi^{*}\frac{\partial\psi}{\partial y}\right),$ (50) $\displaystyle S_{z}$
$\displaystyle=$ $\displaystyle 0.$ (51)

Hence, the energy flow pattern determined by eqn. (41) is given by

$\frac{dy}{dx}=\frac{S_{y}}{S_{x}}=\frac{\left(\psi\frac{\partial\psi^{*}}{\partial
y}-\psi^{*}\frac{\partial\psi}{\partial
y}\right)}{\left(\psi\frac{\partial\psi^{*}}{\partial
x}-\psi^{*}\frac{\partial\psi}{\partial x}\right)}.$ (52)

Since $\rho_{mes}$ drops out of this ratio, the energy flow pattern is the
same for all values of $\lambda(\theta)$. This is an important result of
relevance for the interpretation of empirically determined energy flow
patterns [6].

###  4.2 Optical Vortices

The easy production of light beams with discrete angular momentum in the
laboratory using laser beams has led to novel experiments to manipulate
microparticles [7]. Lasers typically produce Hermite-Gauss beams. Helical
Laguerre-Gauss beams can be readily produced by conversion of these Hermite-
Gauss modes using specially designed holograms. They have helical phase fronts
because the Poynting vector is not parallel to the beam axis but follows a
spiral trajectory, creating ‘optical vortices’ with zero intensity along the
beam axis.

The paraxial approximation of the Helmholtz solutions is the most appropriate
to desctibe these modes. For linearly polarized light the paraxial fields are
given by [8]

$\displaystyle\vec{B}$ $\displaystyle=$
$\displaystyle=ik\left[u\hat{y}+\frac{i}{k}\frac{\partial u}{\partial
y}\hat{z}\right]e^{ikz},$ (53) $\displaystyle\vec{E}$ $\displaystyle=$
$\displaystyle ik\left[u\hat{x}+\frac{i}{k}\frac{\partial u}{\partial
x}\hat{z}\right]e^{ikz},$ (54)

where the propagation is now taken along the $z$ axis, $u$ being the complex
scalar function that describes the distribution of the field amplitude in the
paraxial approximation. In the Lorentz gauge and in cylindrical coordinates,
this function is given by [9]

$\displaystyle u_{pl}(r,\phi,z)$ $\displaystyle=$
$\displaystyle\frac{C}{(1+z^{2}/z^{2}_{R})^{1/2}}\left[\frac{r\sqrt{2}}{w(z)}\right]^{l}L_{p}^{l}\left[\frac{2r^{2}}{w^{2}(z)}\right]$
(55) $\displaystyle\times$ $\displaystyle{\rm
exp}\left[\frac{-r^{2}}{w^{2}(z)}\right]{\rm
exp}\left(\frac{-ikr^{2}z}{2(z^{2}+z_{R}^{2})}\right){\rm
exp}\left(-il\phi\right)$ $\displaystyle\times$ $\displaystyle{\rm
exp}\left[i(2p+l+1){\rm tan}^{-1}\frac{z}{z_{R}}\right],$

where $C$ is a constant, $z_{R}$ is the Rayleigh range, $w(z)$ is the radius
of the beam, $L_{p}^{l}$ is the associated Laguerre polynomial, and the beam
waist is at $z=0$. The time averaged Poynting vector for a beam of unit
amplitude is then given by

$\displaystyle\vec{S}=\mu_{0}^{-1}(\vec{E}\times\vec{B})$ $\displaystyle=$
$\displaystyle\frac{1}{2\mu_{0}}\left[\langle\vec{E}^{*}\times\vec{B}\rangle+\langle\vec{E}\times\vec{B}^{*}\rangle\right]$
(56) $\displaystyle=$ $\displaystyle i\omega\frac{1}{2\mu_{0}}\left(u\nabla
u^{*}-u^{*}\nabla u\right)+\frac{\omega}{\mu_{0}}k|u|^{2}\hat{z}.$

Using the LG mode given by (55) for linearly polarized beams, one can show
that [9]

$\displaystyle\vec{S}=c\left[\frac{rz}{z^{2}+z_{R}^{2}}|u|^{2}\hat{r}+\frac{l}{kr}|u|^{2}\hat{\phi}+|u|^{2}\hat{z}\right]$
(57)

where $|u|^{2}=|u(r,\phi,z)|^{2}$ and the hats denote unit vectors. This shows
that the Poynting vector spirals along the direction of propagation $\hat{z}$.

One can calculate $\rho_{mes}$ from (37), (53), (54) and the quantum potential
is given by

$\displaystyle Q$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{\omega^{2}\epsilon_{0}}\frac{\Delta^{2}|u(r,\phi,z)|/\Delta
B^{2}}{|u(r,\phi,z|}.$ (58)

One can then compute the energy flow lines from eqn. (41) and study the
continuous transtions between the quantum and classical flows.

##  5 Acknowledgement

The author thanks the National Academy of Sciences, India for a grant. He also
specially thanks Anirban Mukherjee for a very helpful discussion and Riddhi
Chatterjee for carefully checking the calculations.

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