X-pattern, i.e. the region inside the red boxes in Fig. 29 (b) and (c) cancel
against the BCJ-pattern. This cancellation does not rely on the substructures
outside the red boxes. Thus the last term of (5.6) should cancel with the
corresponding terms analogue to the second term of (5.6) but $\mathscr{U}_{1}$
is further divided into two subgraphs.

Figure 33: Graph (a) is a typical graph with seven subgraphs, while graph (b)
is the diagram corresponding to (a) with the relative order
$A_{1}^{\prime\prime},A_{4}^{\prime},A_{7},A_{5},A_{2},A_{3},A_{6}$. Unlike
the diagram in Fig. 32, the subcurrents $A_{4}^{\prime}$ and $A_{7}$ are now
adjacent to each other, and the X-pattern of $A_{4}^{\prime}$ provides a
nonvanishing term.

When the cancellations with the above pattern between diagrams corresponding
to distinct cuts are all taken into account, we only need to study the
boundary cases.

#### The boundary cases of the cancellations

(i) The first boundary case is that $\mathscr{U}_{1}$ in (5.6) contains only
one node. In this case, the on-shell condition gives
$k_{\mathscr{U}_{1}}^{2}=0$, and $\mathscr{U}_{1}$ cannot be further divided.
Therefore, the third term in (5.6) itself vanishes but do not cancel with a
further division.

(ii) The second boudary case is that the subcurrent on the right hand of
$A_{1}^{\prime}$ (which contains an X-pattern) in Fig. 30 (b) does not contain
elements of the tree structure
$\mathcal{F}_{4}^{\prime}\oplus\mathcal{F}_{7}$. Then we cannot find a
corresponding Fig. 30 (c) that involves a larger subcurrent whose elements all
belong to $\mathcal{F}_{4}^{\prime}\oplus\mathcal{F}_{7}$. For this situation,
for instance, we encounter the cancellations between diagrams in Fig. 32 which
correspondingly come from the cuts shown by Fig. 31 instead ( In Fig. 31 (c),
$\mathcal{F}_{4}^{\prime}$ plays as the upper block and $\mathcal{F}_{5}$
plays as the lower one of a larger subgraph). When the BCJ relation is applied
to the second subcurrent in Fig. 32 (c), we get a factor
$\phi_{\left[\mathcal{F}^{\prime}_{4},\mathcal{F}_{5}\right]}$. The
contribution of the second term of
$\phi_{\left[\mathcal{F}^{\prime}_{4},\mathcal{F}_{5}\right]}$ cancels out
with the second term of Fig. 32 (b), if (b) is expanded in a similar way with
(5.6). Hence this boundary finally cancels out.

(iii) The third boundary case occurs when the full tree structure
$\mathcal{F}^{\prime}_{4}\oplus\mathcal{F}_{7}$ is not separated by subcurrent
containing elements of other tree (see Fig. 33 (b) for example, which
corresponds to Fig. 33 (a)). For this case, the diagram Fig. 33 (b) cannot all
cancel out. It contributes a nonvanishing term

$\displaystyle\left[\left(\mathcal{C}^{\mathscr{L}_{1}}\phi_{\mathscr{L}_{1}}\right)\left(\mathcal{C}^{\mathcal{F}_{4}^{\prime}}\phi_{\mathcal{F}_{4}^{\prime}}\right)\left(\mathcal{C}^{\mathcal{F}_{7}}\phi_{\mathcal{F}_{7}}\right)\dots\right]{1\over
s_{\mathscr{L}_{1},l}}{1\over s_{\mathscr{L}_{1}A^{\prime}_{4}A_{7},l}}\dots,$
(5.10)

which has a linear propagator $1\over
s_{\mathscr{L}_{1}A^{\prime}_{4}A_{7},l}$ (containing the total momentum of
the nodes in $\mathcal{F}^{\prime}_{4}\oplus\mathcal{F}_{7}$), together with
two subcurrents (and the corresponding coefficents) that are obtained by
cutting $\mathcal{F}^{\prime}_{4}\oplus\mathcal{F}_{7}$ into two connected
subtrees.

###  5.5 From linear to quadratic propagators

Figure 34: The typical diagram with surviving terms, where $A_{1}$, $A_{3}$
are the subcurrents corresponding to the physical subgraphs. The $B_{1L}$ and
$B_{1R}$ together form another type of surviving term, in which the propagator
between them is cut off due to the X-pattern of $B_{1L}$.

We now summarize what we have learnt in the previous subsections and further
demonstrate the general approach for extracting quadratic propagators.

  * •

Given a graph and a set of cuts, if it contains any subgraph with the BCJ-
pattern (e.g. Fig. 29 (a)), one can always find the corresponding graphs with
X-pattern (e.g. Fig. 29 (b)) cancel against it. Therefore, the surviving
graphs cannot involve any subgraph with the BCJ-pattern.

  * •

If a graph $\mathcal{F}_{j}$ contains an X-pattern, it corresponds to three
terms with cutting the linear propagators on the left, right of the current
$\phi_{A_{j}}$, or the off-shell line of $\phi_{A_{j}}$. After cancellations,
the remaining term, as shown by Fig. 34, can only come from the case with
cutting the right propagator between two subcurrents $\phi_{A_{j}}$ and a
subcurrent $\phi_{A_{k}}$ which contains the highest-weight node above
$\mathcal{F}_{j}$. This surviving case is resulted by the boundary case (iii),
accompanied by a coefficient
${\mathcal{C}}^{\mathcal{F}_{j}}\cdot{\mathcal{C}}^{\mathcal{F}_{k}}$.

  * •

When the BCJ-pattern and X-pattern are constructed, there may still exist
remaining physical graphs and additional spurious graphs. These graphs
together form a physical graph with an overall factor
$\mathcal{C}^{\mathcal{F}_{k}}\cdot k_{A_{j}}$ (where $\mathcal{F}_{k}$ is the
subgraph above $\mathcal{F}_{j}$) extracted out.

With the above observations, we further rearrange the summations (5.2) back to
(3.2) when all cancellations between the BCJ-pattern and the X-pattern have
been completed. Then (3.2) turns into

$\displaystyle\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{1}\,...\,\boldsymbol{\sigma}_{I}=\boldsymbol{\sigma}\\\
H_{1}\cup...\cup H_{I}=H\end{subarray}}\,{{1\over
l^{2}}}\,J\,\left[A_{1}\right]\,{{1\over
s_{A_{1},l}}}J\,\left[A_{2}\right]{{1\over
s_{A_{1}A_{2},l}}}J\,\left[A_{3}\right]\times...\times{1\over
s_{A_{1}A_{2}...A_{I-1},l}}J\,\left[A_{I}\right]+\text{cyc}[A_{1},A_{2},...,A_{I}].$

In the above expression, each $A_{j}$ ($j=1,...,I$) is defined by the elements
of $\boldsymbol{\sigma}_{j}\cup H_{j}$, where $\boldsymbol{\sigma}_{j}$ is an
ordered subset of scalars and $H_{j}$ is a subset of gluons. Either
$\boldsymbol{\sigma}_{j}$ or $H_{j}$ may be empty, but
$\boldsymbol{\sigma}_{j}\cup H_{j}$ has at least one element. Each
$J\,\left[A_{j}\right]$ $(j=1,...,I)$ is given by

$\displaystyle J\,\left[A_{j}\right]$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathcal{F}_{j}}\sum\limits_{\shuffle}\mathcal{C}^{\mathcal{F}_{j}}\phi_{\boldsymbol{\sigma}_{j}\shuffle\boldsymbol{\rho}^{\mathcal{F}_{j}}|\widetilde{A}_{j}}\,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{if
$A_{j}$ contains scalars\,}),$ (5.12) $\displaystyle J\,\left[A_{j}\right]$
$\displaystyle=$
$\displaystyle\sum\limits_{\mathcal{F}_{j}}\sum\limits_{\shuffle}\mathcal{C}^{\mathcal{F}_{j}}\cdot\left(l+k_{A_{1}}+...+k_{A_{j-1}}\right)\phi_{\boldsymbol{\rho}^{\mathcal{F}_{j}}|\widetilde{A}_{j}}$
(5.13)
$\displaystyle+\sum\limits_{\mathcal{F}_{jL},\mathcal{F}_{jR}}\,\mathcal{C}^{\mathcal{F}_{jL}}\cdot\mathcal{C}^{\mathcal{F}_{jR}}\,\left[\,\sum\limits_{\boldsymbol{\rho}^{\mathcal{F}_{jL}},\boldsymbol{\rho}^{\mathcal{F}_{jR}}}\,\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jL}}|\widetilde{A}_{jL}}\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jR}}|\widetilde{A}_{jR}}\right]\,(\text{if
$A_{j}$ does not contain scalar\,}),$

in which $\mathcal{F}_{j}$ denotes all graphs generated by the refined graphic
rule, with the reference order that inherits the one for the full graph. The
ordered set $\boldsymbol{\sigma}_{j}$ plays as the scalar trace and $H_{j}$
plays as the gluon set. The
$\boldsymbol{\sigma}_{j}\shuffle\boldsymbol{\rho}^{\mathcal{F}_{j}}$ are
possible permutations defined by the subgraph $\mathcal{F}_{j}$, where the
off-shell line of the BS current is considered as the last node. The
coefficients $\mathcal{C}^{\mathcal{F}_{j}}$ in eq. (5.12) and eq. (5.13)
denote the kinematic factors characterized by the subgraphs
${\mathcal{F}_{j}}$. Note that in eq. (5.12), the graph $\mathcal{F}_{j}$ is a
graph for single-trace YMS current with an off-shell scalar. In eq. (5.13),
the graph $\mathcal{F}_{j}$ is a graph for YM current with an off-shell gluon,
hence $\left[\mathcal{C}^{\mathcal{F}_{j}}\right]^{\mu}$ has a Lorentz index.
If the subset $A_{j}$ does not contain any scalar, an additional term (the
second term of eq. (5.13)) should also be taken into account: we split
$A_{j}=H_{j}$ into two subsets $A_{jL}$ and $A_{jR}$, the latter contains the
highest-weight element in $A_{j}$. Then we sum over all possible graphs
$\mathcal{F}_{jL}$, $\mathcal{F}_{jR}$ constructed by elements in $A_{jL}$ and
$A_{jR}$, respectively. The coefficient
$\mathcal{C}^{\mathcal{F}_{jL}}\cdot\mathcal{C}^{\mathcal{F}_{jR}}$ is given
by the contraction of $\left[\mathcal{C}^{\mathcal{F}_{jL}}\right]^{\mu}$ and
$\left[\mathcal{C}^{\mathcal{F}_{jR}}\right]^{\mu}$. For given
$\mathcal{F}_{jL}$, $\mathcal{F}_{jR}$, we sum the product of two BS
subcurrents $\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jL}}|\widetilde{A}_{jL}}$,
$\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jR}}|\widetilde{A}_{jR}}$ with
permutations $\boldsymbol{\rho}^{\mathcal{F}_{jL}}$,
$\boldsymbol{\rho}^{\mathcal{F}_{jR}}$ established by the two subgraphs. The
$\text{cyc}[A_{1},A_{2},...,A_{I}]$ in eq. (5.5) means that we should sum all
terms with cyclic permutations $A_{1}$, $A_{2}$, …, $A_{I}$. This cyclic sum
comes from the sum over cyclic permutations of $\boldsymbol{\sigma}$ and
$\boldsymbol{\gamma}$ in (3.2). For example, if $I=3$, $A_{1}$, $A_{2}$ and
$A_{3}$ become the subsets of elements in $\sigma_{1}\cup H_{1}$,
$\sigma_{1}\cup H_{2}$ and $\sigma_{3}\cup H_{3}$. Thus
$\boldsymbol{\sigma}=\boldsymbol{\sigma}_{1}\boldsymbol{\sigma}_{2}\boldsymbol{\sigma}_{3}$,
$\boldsymbol{\gamma}=\widetilde{A}_{1}\widetilde{A}_{2}\widetilde{A}_{3}$. The
cyclic sums over $\boldsymbol{\sigma}$ and $\boldsymbol{\gamma}$ in (3.2) then
allow those terms with cyclic permutations of both $A_{i}$ and
$\widetilde{A}_{i}$ ($i=1,2,3$) (hence the cyclic permutations of
$J[A_{1}],J[A_{2}],J[A_{3}]$). According to eq. (2.35), the expression (5.5)
for single-trace YMS integrand turns into

$\displaystyle\frac{1}{l^{2}}\int\text{d}\mu_{\,n+2}^{\,\text{tree}}I_{\,\text{L}}^{\,\text{1-loop}}I_{\,\text{R}}^{\,\text{1-loop}}$
$\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{1}\,...\,\boldsymbol{\sigma}_{I}=\boldsymbol{\sigma}\\\
H_{1}\cup...\cup H_{I}=H\end{subarray}}\,{{1\over
l^{2}}}\,J\,\left[A_{1}\right]\,{{1\over(l+k_{A_{1}})^{2}}}J\,\left[A_{2}\right]{{1\over(l+k_{A_{1}}+k_{A_{2}})^{2}}}J\,\left[A_{3}\right]$
(5.14)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times...\times{1\over(l+k_{A_{1}}+k_{A_{2}}+...+k_{A_{I-1}})^{2}}J\,\left[A_{I}\right]$
$\displaystyle=\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{1}\,...\,\boldsymbol{\sigma}_{I}=\boldsymbol{\sigma}\\\
H_{1}\cup...\cup
H_{I}=H\end{subarray}}\,gon\left(A_{1},A_{2},...,A_{I}\right)\prod\limits_{j=1}^{I}J\,\left[A_{j}\right].$

in which, all linear propagators have been converted into quadratic ones. _It
is worth pointing out that the boundary case when there are only external
gluons but no external scalars is also given by eq. ( 5.14). But in this case,
the $J\,\left[A_{j}\right]$’s are only given by (5.13)._

##  6 Quadratic propagators from refined graphic rule: explicit examples

Figure 35: The cancellations between the X-pattern and BCJ-pattern of graphs
Fig. 41 (a1) and (a2), where the surviving part is boxed with the dependence
on the gluons momenta $-(k_{r}\cdot k_{q})$ shown explicitly Figure 36: The
cancellation route maps induced by Fig. 41 (b1), where (b10C1) is the only
term that survives after considering all cancellations, including those
between the remaining parts of (b3A), (b10A) and terms induced by other graphs
in Fig. 41.

In this section, we provide a calculation for a part of single-trace
amplitudes with four gluons: the contributions with two
$\epsilon\cdot\epsilon$’s. This example illustrates the general approach in
the previous section explicitly. Other contributions of four gluon case and in
general an arbitrary number of gluons follow from an analogue discussion, and
one finally arrive the quadratic propagator expression (5.14).

Figure 37: All diagrams we obtained, after considering the cancellations of
all graphs in Fig. 41. The boxed diagrams are already in the proper form,
while those unboxed diagrams can further be collected. For example, diagrams
(A1), (A6), and (A7), which have the same structure but different
coefficients, can be collected together to provide the coefficient
$-k_{r}\cdot X_{r}$. Figure 38: The diagrams coming from collecting the
unboxed diagrams in Fig. 38, which, together with the boxed diagrams in Fig.
38, are all surviving terms of Fig. 41.

In this section, we show the cancellation route maps explicitly by starting
with the maximal cut of gluons (i.e. the cut such that each gluon belongs to a
one-element subset). In general, the number of cut dashed lines between
scalars does not affect our discussion, thus possible cut dashed lines are not
explicitly displayed.

Cancellations induced by Fig. 41 (a1): We begin with the graph Fig. 41 (a)
with four cut gluon lines, i.e. Fig. 41 (a1). In Fig. 41 (a1), the node $p$
itself is a subgraph, and is connected to scalars via a type-3 line. On the
Feynman diagram side with linear propagators, there are two distinct
situations: (i). $p$ is adjacent to scalar subset that is connected to it,
(ii). $p$ is seperated from the scalar subset, by another scalar subcurrent.
For the latter case, one can always find other graphs in which $p$ is rooted
at other subsets of scalars, and the coefficients of all these contributions
are collected as $-k_{p}\cdot X_{p}$, where $X_{p}$ is the sum of momenta of
all scalars located at the LHS of $p$. Then we can regard the current on the
left hand side of $p$ as the new current that is connected to $p$ instead and
turn to the case (i).

To handle with the case (i), we find that the graph Fig. 41 (a1) has an
X-pattern, thus it can be regarded as the graph Fig. 29 (b), where $p$ plays
as the $\mathcal{F}_{1}^{\prime}$ and the subgraph below $p$ is considered as
the graph $\mathcal{F}_{1}^{\prime\prime}$. Therefore, Fig. 41 (a1)
contributes graphs with cutting the left linear propagator Fig. 35 (a1A) and
cutting the right linear propagator Fig. 35 (a1B1), (a1B2). The former case
cancels out with a BCJ-pattern Fig. 35 (a2) which comes from Fig. 41 (a2),
while the latter case, if the $p$ and $q$ are separated by some scalar
subcurrents, has to cancel with another BCJ-pattern Fig. 35 (a1’). The graph
Fig. 35 (a1B2) where $p$ and $q$ are adjacent to each other is the surviving
one in this step cancellation. Nevertheless, one can further carry out
cancellations between distinct positions of $r$, $s$ by considering $p$ and
$q$ as scalar subcurrents.

Cancellations induced by Fig. 41 (b1): As discussed in the previous part, the
graph Fig. 41 (b1), which also contains X-pattern, has two contributions
corresponding to the cases with the left or the right linear propagator
cut.the part where the left linear propagator to $q$ is cut (as shown by Fig.
36 (b1A)) cancels against the graph with a BCJ-pattern Fig. 36 (b2). The
remaining term is the part with cutting the linear propagator to the right of
$q$, which are further displayed by Fig. 36 (b1B1) and (b1B2) according to
whether $q$, $p$ are separated by scalar subcurrents or not. The graph Fig. 36
(b1B1) cancels with the BCJ-pattern Fig. 36 (b2’). The graph Fig. 36 (b1B2)
gets cancellation from Fig. 36 (b3A) which comes from a part of contribution
of Fig. 41 (b3) and (c3) with an X-pattern.

According to the X-pattern, the remaining contributions of Fig. 41 (b3) and
(c3) are the graph where the linear propagator to the left or the right of the
subcurrent containing $p,q$ is cut, presented by Fig. 36 (b3B) and Fig. 36
(b3C1), Fig. 36 (b3C2), respectively. Again, the Fig. 36 (b3B) with the left
cut propagator cancels with a BCJ-pattern Fig. 36 (b11) which comes from Fig.
42 (b11) and (c11). One can find Fig. 36 (b10A) (X-pattern with a larger
subcurrent coming from Fig. 41 (b10), (c10) and (d10)) and Fig. 36 (b11’) (a
BCJ-pattern coming from Fig. 41 (b11) and (c11)) which respectively cancel the
Fig. 36 (b3C1) and Fig. 36 (b3C2).

Once the above cancellations have been accomplished, other contributions of
Fig. 41 (b10), (c10) and (d10) have been introduced, i.e., Fig. 36 (b10B) and
(b10C1), (b10C2). The first has the X-pattern with cutting the left linear
propagator, while the last two with the right linear propagator cut. Following
a similar cancellation process which has been applied before, the Fig. 36
(b10B) and (b10C2) cancel with the BCJ-patterns Fig. 36 (b15) and (b15’) that
come from Fig. 43 (b15), (c15) and (d15). So far, we have no further
cancellations induced by the graph Fig. 41 (b1) and the surviving graphs are
(i) a part of Fig. 36 (b10A) and (ii). a part of Fig. 36 (b3A), and (iii) the
graph Fig. 36 (b10C1). The (i) is cancelled out when considering the
cancellation route induced from (b4), (c1), (d1) while (ii) is cancelled out
in the route induced from (c1). Therefore, the only surviving term in this
route map is Fig. 36 (b10C1).

Similar cancellations are induced by other graphs: For each graph with cutting
four gluon lines, one induces a cancellation route map. In each route map,
there is no surviving BCJ-pattern and no X-pattern with cutting the left
linear propagator or the propagator of the tree subcurrent. Hence, the
surviving terms are those including (i). the X-pattern with the right
propagator cut between two subcurrents containing only gluons (the right
subcurrent involves the highest-weight gluon of the two subcurrents), and/or
(ii). subcurrents with a physical subgraph under the same reduced reference
order, as shown by Fig. 37. Once four-gluon-line cut has been treated, we turn
to the remaining graphs with fewer cuts and do the same reduction. When the
cancellations induced from all graphs are considered, we obtain graphs in Fig.
37. Those boxed diagrams in Fig. 37, are already in the proper form. The
unboxed diagrams which have distinct $k\cdot k$ coefficients but the same
structure, can further be collected together and produce a factor, for
example, for Fig. 37 (A1), (A2) and (A7), the factor is $-k_{r}\cdot X_{r}$.
This factor allows one to get a proper form such that corresponding
subcurrents just contract with the total momentum on the left. Finally, all
the remaining contributions are the boxed diagrams in Fig. 37 and Fig. 38.
Each boxed diagram contains substructures with a physical graph or a surviving
X-pattern with the right propagator cut, such a substructure is just a part of
the $J[A_{j}]$ in eq. (5.12) and eq. (5.13). When we sum over cyclic
permutations of $\\{A_{j}\\}$, according to eq. (2.35), the linear propagators
are transformed into quadratic ones. For instance, the diagram Fig. 38 (C), if
contains only subsets $A_{1}=\mathcal{L}$, $A_{2}=\\{p,q\\}$ and
$A_{3}=\\{r,s\\}$ and does not contain other subset of scalars, gives rise

$\displaystyle{1\over
l^{2}}\,\phi_{\mathcal{L}}\,{1\over(l+k_{\mathcal{L}})^{2}}\,\left[(\epsilon_{p}\cdot\epsilon_{q})\phi_{p}\phi_{q}\right]\,{1\over(l+k_{\mathcal{L}}+k_{p}+k_{q})^{2}}\,\left[(\epsilon_{r}\cdot\epsilon_{s})(-k_{r}\cdot
X_{r})\phi_{rs}\right],$ (6.1)

where $X^{\mu}_{r}$ is the sum of $l^{\mu}$ and the external momenta between
the loop propagator ${1\over l^{2}}$ and the subcurrent $\phi_{rs}$,
explicitly, $X^{\mu}_{r}=(l+k_{\mathcal{L}}+k_{p}+k_{q})^{\mu}$.

Figure 39: Vertices in the Feynman rule for single-trace YMS amplitudes (see
Bern:1999bx ) This Feynman rule evaluates the color-ordered YMS amplitude of
YM and color-scalar. When the color factors for color-scalars are further
extracted out, the kinematic part becomes the YMS amplitudes discussed in this
paper.

Cancellations induced by other graphs with two $\epsilon\cdot\epsilon$’s as
well as the graphs with one and no $\epsilon\cdot\epsilon$ follow a similar
discussion. When putting all terms with respect to a same set of loop
propagators, we get a term in eq. (5.14). Thus eq. (5.14) is finally obtained
by collecting contributions corresponding to distinct sets of loop
propagators.

##  7 Traditional Feynman diagrams and the extension to multi-trace YMS and YM

In this section, we reveal the relationship between the main result (5.14)
that was derived from one-loop CHY formula, and the direct analysis from
Feynman diagrams. We further extend these results straightforwardly to multi-
trace YMS amplitudes and pure YM amplitudes.

###  7.1 The relationship with traditional Feynman diagrams

Now we demonstrate the explicit relationship between the quadratic propagator
form (5.14) and the Feynman diagrams. To achieve this, we display the
expressions of vertices in traditional Feynman rule for single-trace YMS in
Fig. 39 and then package the Feynman diagrams by attaching tree-level BG
currents to the loop with quadratic propagators, via vertices in Fig. 39. The
sum of all Feynman diagrams become the sum of diagrams with structures like
Fig. 40 (a). The BG currents in general can be planted into the quadratic-
propagator scalar loop via three types of vertices, as shown by Fig. 40 (b),
(c), (d). Each $J^{\mu}$ in Fig. 40 (b) and (c) is a BG current for gluons,
which has been proven (by Feynman diagrams) to satisfy the off-shell
decomposition formula Wu:2021exa  (also see Lee:2015upy ; Bridges:2019siz ):

$\displaystyle J^{\mu}=\widetilde{J}^{\mu}+K^{\mu}+L^{\mu},$ (7.1)

Figure 40: Traditional Feynman diagrams for single-trace YMS in general can be
packaged into (a), where tree structures are collected as BG currents which
are further attached to the loop with quadratic propagators via the vertices
Fig. 39 (c), (d) and/or (e). Thus there are three types of such substructures,
as shown by (b), (c) and (d) (the vertices Fig. 39 (c), (d) and (e) are
respectively denoted by $V_{3}$, $V^{\prime}_{3}$ and $V_{4}$), where
$J^{\mu}[A_{j}]$’s in (b) and (c) are pure gluon BG currents, while $J[A_{j}]$
in (d) is a YMS BG current with both scalar and gluon external lines.

in which the first term $\widetilde{J}^{\mu}$ can be decomposed according to
the refined graphic rule555Here we make some clarifications about the result
in Wu:2021exa : (i). In Wu:2021exa , the reference order was fixed as the
inverse of the right permutation of the BS amplitude. However, this result can
be generalized to other reference orders with changing the particular
expressions of the $K$ term and $L$ term. (ii). In Wu:2021exa , the expansion
of $\widetilde{J}$ is expressed by summing over permutations, but in this
paper, we sum over graphs. In fact they are equivalent to each other as
pointed out in Hou:2018bwm . (iii). In Wu:2021exa , there are two equivalent
decompositions of $\widetilde{J}$: (a) The first is choosing the off-shell
line as the last element, and an on-shell line as the first. Assuming that
there are $m$ external on-shell particles, this is an $(m-1)!$-decomposition
and the coefficients were mentioned as type-A numerators. (b). The second is
to choose the off-shell element as the first one but one does not fix the
last. This is an $m!$-decomposition and the coefficients are mentioned as
type-B numerators in Wu:2021exa . In this work, the $\widetilde{J}$ in eq.
(7.1) is considered as the second decomposition. (iv). In Wu:2021exa , each
momentum $k^{\mu}$ appeared as $2k^{\mu}$, this factor $2$ plays as an overall
factor and can be absorbed, thus does not affect our discussion. . The second
and the third contain substructures which are obtained via replacing
subcurrents by objects proportional to the total momentum of these
subcurrents. Note that the cubic vertex Fig. 39 (c) with two scalar lines and
one gluon line is proportional to $\delta^{ab}2k_{1}^{\mu}$, when the Ward
identity for tree-level current is considered. The substructure Fig. 40 (a),
together with $\widetilde{J}^{\mu}$ produces (upto a factor 2)

$\displaystyle\widetilde{J}\left[A_{j}\right]\cdot\left(l+k_{A_{1}}+...+k_{A_{j-1}}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{\mathcal{F}_{j}}\sum\limits_{\boldsymbol{\rho}^{\mathcal{F}_{j}}}\mathcal{C}^{\mathcal{F}_{j}}\cdot\left(l+k_{A_{1}}+...+k_{A_{j-1}}\right)\phi_{\boldsymbol{\rho}^{\mathcal{F}_{j}}|\widetilde{A}_{j}},$
(7.2)

which is nothing but the first term of eq. (5.13) in the case that the set
$A_{j}$ is a pure gluon set. In Fig. 40 (b), the four-point vertex, together
with the $\widetilde{J}^{\mu}$ contributes a

$\displaystyle\widetilde{J}[A_{jL}]\cdot\widetilde{J}[A_{jR}]=\sum\limits_{\mathcal{F}_{jL},\mathcal{F}_{jR}}\,\left[\mathcal{C}^{\mathcal{F}_{jL}}\cdot\mathcal{C}^{\mathcal{F}_{jR}}\right]\,\sum\limits_{\boldsymbol{\rho}^{\mathcal{F}_{jL}},\boldsymbol{\rho}^{\mathcal{F}_{jR}}}\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jL}}|\widetilde{A}_{jL}}\phi_{\boldsymbol{\rho}^{\mathcal{F}_{jR}}|\widetilde{A}_{jR}},$
(7.3)

where $\mathcal{F}_{jL}$ and $\mathcal{F}_{jR}$ are the graphs constructed by
nodes in $A_{L}$ and $A_{R}$ respectively. When the graphs $\mathcal{F}_{jL}$
and $\mathcal{F}_{jR}$ are produced according to the reference orders
resulting from the one in the expansion of one-loop CHY formula, and all
possible splittings of the ordered subset $A_{jL}$, $A_{jR}$ are considered,
we get the second term in eq. (5.13) when fixing the reference order of gluons
as the same with the relative permutations in right half integrand666A
subtlety is that the Feynman rule is the color-dressed one. One can always
absorb $\delta^{ab}$ into the color factor, then there exist two relative
orders of $A_{jL}$ and $A_{jR}$ corresponding to different color orderings. If
the highest-weight node belongs to the right set (when the reference order is
chosen as the relative order of gluons in the right half integrand, this is
always guaranteed), it agrees with the second term in eq. (5.13).
Nevertheless, if the highest-weight node in $A_{jL}\cup A_{jR}$ belongs to the
left set, the Feynman diagram exists but there is not a direct correspondence
in eq. (5.13). This is because the reference order in eq. (5.13) breaks the
exchanging symmetry between gluons. In fact, one can expect that distinct
reference orders are always related via gernalized BCJ identities
(Boels:2011tp ; Du:2012mt ). In order to provide a symmetric construction, we
can take the average of eq. (5.13) for all reference orders..

Although the decomposition formula of the YMS BG current has not been
explicitly given, it is natural to expect that the current mixing scalars and
gluons can also split into the part decomposing according to the graphic rule
and another part which has to vanish in the on-shell limit. The former part is
in general constructed by attaching the $\widetilde{J}$ part of a fewer-gluon
current to scalar trees. When substituted into Fig. 40 (d), this part
precisely matches the first term of eq. (5.13) in the case of
$\boldsymbol{\sigma}_{j}\neq\emptyset$. The latter part, together with the
contributions of the last two terms in eq. (7.1) have to cancel out due to
Ward identities, for there exist substructures where a Berends-Giele
subcurrent is replaced by its total momentum777The detail of this statement
will be presented in a coming work.. Hence, the Feynman diagrams precisely
match the one eq. (5.14) obtained from one-loop CHY.

###  7.2 The extensions to multi-trace YMS and YM

Although all the previous discussions are carried around single-trace YMS
amplitudes, it is straightforward to generalize them to multi-trace YMS
amplitudes with a pure scalar loop. This is because the multi-trace YMS
amplitudes that are derived from tree-level CHY formula can be obtained via
introducing another type of components: components with a gluon trace as
stated in section 2. Such a component has an equal status with the components
containing a type-1 line. Therefore, one just includes these new components in
the previous discussions, and finally also arrives the conclusion eq. (5.13),
but in this case, the graphs $\mathcal{F}_{jL}$, $\mathcal{F}_{jR}$ referred
in eq. (5.13) can be those with both gluons and scalar traces 888 On the
Feynman diagram side, this feature naturally implies the four-scalar vertex
for multi-trace YMS amplitudes.. Note that the multi-trace case discussed
above is only the case where all loop propagators are scalar ones and the two
particles $\pm$ with the forward limit are adjacent to each other in the
$(n+2)$-point tree-level half integrand. As classified in Porkert:2022efy ,
according to distinct orders of coupling constants, there still exist (i).
half integrands which contribute only scalar loop propagators but the
particles $\pm$ are not adjacent to each other, and (ii). the half integrands
which contribute gluon propagators on the loop. The former case is related to
the results in the current paper via a KK relation at one-loop level (see
appendix B), while we leave a detailed study of the latter case in future
work.

For YM amplitudes, we have to expand the reduced Pfaffian (see table 2) with
$n+2$ external particles in terms of tree-level PT factors and then take the
forward limit. This can be achieved via eq. (2.3.3) where each term is
accompanied by a Pfaffian for YMS. More specifically, apart from the prefactor
$\text{Tr}[F_{i_{1}}\cdot F_{i_{2}}\cdot...\cdot F_{i_{l}}]$, each term in eq.
(2.3.3) is a YMS half integrand with the scalar trace
$+,i_{1},i_{2},...,i_{l},-$. Hence, together with the other half integrand,
which is expressed as an $(n+2)$-point PT factor where the $\pm$ are
considered as the two ends and the cyclic permutations over other $n$ elements
are summed over, such a term will produce

$\displaystyle(-1)^{l}\,\text{Tr}[F_{i_{1}}\cdot F_{i_{2}}\cdot...\cdot
F_{i_{l}}]\,\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{1}\,...\,\boldsymbol{\sigma}_{I}=i_{1}i_{2}...i_{l}\\\
H_{1}\cup...\cup
H_{I}=\\{1,...,n\\}\setminus\\{i_{1},...,i_{l}\\}\end{subarray}}\,gon\left(A_{1},A_{2},...,A_{I}\right)\prod\limits_{j=1}^{I}J\,\left[A_{j}\right],$
(7.4)

where the cyclic symmetry of $\text{Tr}[F_{i_{1}}\cdot F_{i_{2}}\cdot...\cdot
F_{i_{l}}]$ has been applied. The $J\,\left[A_{j}\right]$ has been defined in
eq. (5.12) and eq. (5.13), for single-trace YMS amplitudes with $l$ scalars.
If the set $\\{i_{1},...,i_{l}\\}$ is empty, we do not have the
$\text{Tr}[F_{i_{1}}\cdot F_{i_{2}}\cdot...\cdot F_{i_{l}}]$ and the extra
fator $(D-2)$ in eq. (2.3.3) should be dressed. For this special case, the
$J\,\left[A_{j}\right]$’s are those defined in eq. (5.13) which contains only
external gluons. When all terms in eq. (2.3.3) are collected, one expresses
the one-loop YM amplitudes in terms of expressions with quadratic propagators

$\displaystyle I_{\,\text{YM}}^{\,\text{1-loop}}$ $\displaystyle=$
$\displaystyle(D-2)\sum\limits_{A_{1}\cup...\cup
A_{I}=\\{1,...,n\\}\setminus\\{i_{1},...,i_{l}\\}}gon\left(A_{1},A_{2},...,A_{I}\right)\prod\limits_{j=1}^{I}J\left[A_{j}\right]$
$\displaystyle+$
$\displaystyle\sum\limits_{l=2}^{n}(-1)^{l}\sum\limits_{\\{i_{1},i_{2},...,i_{l}\\}\in\text{S}_{l}\setminus\text{Z}_{l}}\text{Tr}[F_{i_{1}}\cdot
F_{i_{2}}\cdot...\cdot
F_{i_{l}}]\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{1}\,...\,\boldsymbol{\sigma}_{I}=i_{1}i_{2}...i_{l}\\\
H_{1}\cup...\cup
H_{I}=\\{1,...,n\\}\setminus\\{i_{1},...,i_{l}\\}\end{subarray}}gon\left(A_{1},A_{2},...,A_{I}\right)\prod\limits_{j=1}^{I}J\left[A_{j}\right],$

where each $A_{j}$ on the first line and each $\boldsymbol{\sigma}_{j}\cup
H_{j}=A_{j}$ on the second line can at least contain one element. Hence the
$I$ in each term runs from $1$ to $n$ (the total number of external
particles).

##  8 Conclusions and further discussions

In this paper, we provided a general approach to converting the linear
propagators which exist in the one-loop Cachazo-He-Yuan (CHY) formula
Cachazo:2013gna ; Cachazo:2013hca ; Cachazo:2013iea ; Cachazo:2014nsa ;
Cachazo:2014xea  into quadratic propagators which come from traditional
Feynman diagrams. We expanded $n$-point one-loop CHY half integrands for Yang-
Mills-Scalar (YMS) in terms of $(n+2)$-point Parke-Taylor (PT) factors with
forward limits, according to the tree-level refined graphic rule. Then
established correspondence between cuts of the graphs and the Feynman diagrams
with linear propagators. Furthermore, graphs in each family in a given cut
were shown to be constructed by refined graphic rule. After summing over
families and cuts, we found that the subgraphs can be collected as Bern-
Carrasco-Johansson (BCJ)-pattern or X-pattern. When the cancellations between
these patterns were accomplished and the cyclic sums were considered, we
finally got an expression with only quadratic propagators. The formula with
quadratic propagators has further been shown to match the traditional Feynman
diagrams in YMS. The quadratic form reduced from CHY formula of multi-trace
YMS amplitudes and pure Yang-Mills (YM) amplitudes were obtained via the
single-trace YMS. Although, the starting point of this work is the one-loop
CHY formula, the full discussion essentially relies only on the forward limit.
Therefore, the discussion in this paper may be directly extended to other
approaches which are based on forward limit. Moreover, since the quadratic
form of the pure YM amplitudes has been obtained, it is straightforward to
generalize the results to super-Yang-Mills theories and also the amplitudes
when fermions circulate in the loop and the external particles are gluons.

There are still some related problems that deserve further study. The first
one is to implement the conversion from linear to quadratic propagators for
Einstein-Yang-Mills (EYM) and gravity (GR) theories, which have more
complicated CHY half integrands. Being different from YMS and YM, they do not
have a half integrand that can be directly expressed as a scalar PT factor
with a cyclic symmetry Feng:2022wee , which means that the contribution of the
tadpole diagrams of EYM and GR theories needs to be revisited and cannot be
straightforwardly cancelled out by U(1)-decoupling. The second is how to
expand the CHY half integrands into combinations of tensorial PT factors in a
generic way. The third is, we can analyze the algebraic structures and their
connection with one-loop BCJ numerators Bern:2008qj ; Bern:2010ue
corresponding to linear propagators, which may be helpful in constructing one-
loop BCJ numerators corresponding to quadratic propagators (see e.g.
Bern:2013yya ; Chiodaroli:2013upa ; Johansson:2014zca ; Berg:2016fui ;
Edison:2022jln ). The relations between BCJ numerators in various theories at
tree-level Dong:2021qai ; Cao:2022vou  may further help one to extend the
discussions to one-loop amplitudes in different theories. Last but not least,
although we have already analyzed the relationship between the result in the
current paper and Feynman diagrams for YMS, it is still worth working out the
full connection between the YM results and the Feynman diagrams in YM.

## Acknowledgments

We would like to thank Bo Feng, Song He, Yong Zhang and Kang Zhou for helpful
suggestions and comments on the draft. We also thank Chih-Hao Fu and Yihong
Wang for helpful discussions on kinematic algebra. We thank Yixiao Tao, Xinhan
Tong and Konglong Wu for helpful discussions on the decomposition of Berends-
Giele currents in YMS. The authors acknowledge the organizers of “3rd seminar
on quantum field theory and its applications” (2023, Bejing, China). YD would
like to thank the organizers of “4th seminar on field theory and string theory
in China” (2023, Nanjing, China) as well as Fudan University, Tianjin
University for kind invitation. CX would like to thank the organizers of “2023
Frontier Summer School on String Theory, Field Theory, and Holography” (2023,
Nanjing, China). This work is supported by NSFC under Grant No. 11875206.

Figure 41: The graphs (a), (b), (d), (e) and (f) are the physical graphs
constructed by the refined graphic rule with the reference order
$\\{p,q,r,s\\}$, while the graph (c) is the spurious graph we added to get the
BCJ-pattern and X-pattern. A part of cut graphs of these six graphs are given
here, and the remaining cut graphs with fewer cuts are shown by Fig. 42 and
Fig. 43. Figure 42: The two-cut graphs for the four-gluon case with two type-1
lines Figure 43: The one-cut graphs for the four-gluon case with two type-1
lines

##  Appendix A All cuts in the four gluon example

All possible cuts for the example shown in section 6 are presented in Fig. 41,
Fig. 42 and Fig. 43.

##  Appendix B KK relation for one-loop half integrands

As pointed in Porkert:2022efy , there are three types of one-loop CHY half
integrands
$I_{\,\text{L}}^{\,\text{1-loop}}\big{(}\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G}\big{)}$
in the YMS theory which depend on different orders in the couplings $g$ and
$\lambda$, specifically

$\displaystyle
I_{\,\text{L}}^{\,\text{1-loop}}\big{(}\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G}\big{)}|_{g^{n}\lambda^{r-2m+4}}$
$\displaystyle=$ $\displaystyle\sum\limits_{1\leq i<j\leq
m}\sum\limits_{\begin{subarray}{c}Q\in\text{cyc}\,\boldsymbol{\sigma}_{i}\\\
R\in\text{cyc}\,\boldsymbol{\sigma}_{j}\end{subarray}}I_{\,\text{L}}^{\text{tree}}(+,Q,-,R;\boldsymbol{\sigma}_{1};...;\cancel{\boldsymbol{\sigma}_{i}};...;\cancel{\boldsymbol{\sigma}_{j}};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G})+(+\leftrightarrow-)$
$\displaystyle
I_{\,\text{L}}^{\,\text{1-loop}}\big{(}\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G}\big{)}|_{g^{n}\lambda^{r-2m+2}}$
$\displaystyle=$ $\displaystyle
N\sum\limits_{j=1}^{m}\,\sum\limits_{Q\in\text{cyc}\,\boldsymbol{\sigma}_{i}}I_{\,\text{L}}^{\text{tree}}(+,Q,-;\boldsymbol{\sigma}_{1};...;\cancel{\boldsymbol{\sigma}_{i}};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G})+(+\leftrightarrow-)$
$\displaystyle
I_{\,\text{L}}^{\,\text{1-loop}}\big{(}\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G}\big{)}|_{g^{n}\lambda^{r-2m}}$
$\displaystyle=$ $\displaystyle
N\,c_{2}\,I_{\,\text{L}}^{\text{tree}}(\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m};+,-||\boldsymbol{G})+I_{\,\text{L}}^{\text{tree}}(\boldsymbol{\sigma}_{1};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G},+,-)$
(B.1)
$\displaystyle~{}~{}~{}+\sum\limits_{j=1}^{m}\Bigl{\\{}\sum\limits_{\begin{subarray}{c}\boldsymbol{\sigma}_{j}=QR\\\
|Q|,|R|\neq
0\end{subarray}}I_{\,\text{L}}^{\text{tree}}(Q,+;R,-;\boldsymbol{\sigma}_{1};...;\cancel{\boldsymbol{\sigma}_{j}};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G})+\text{cyc}\,\,\boldsymbol{\sigma}_{j}\Bigr{\\}},$

where $N$ refered to the $U(N)$ or $SU(N)$ group for the color scalar, while
the coefficient $c_{2}$ is $N$ for $U(N)$ group and ${{N^{2}-1\over N}}$ for
$SU(N)$ group. The sets $Q$ and $R$ are two ordered sets of scalars, the set
$\boldsymbol{G}$ denotes the gluon set, and $\cancel{\boldsymbol{\sigma}_{i}}$
represents that the $i$-th trace $\boldsymbol{\sigma}_{i}$ is removed. The
number of total particles and scalars are supposed to be $n$ and $r$,
respectively. In the above expression, the RHS of the first and the second
equations as well as the first term on the RHS of the third equation
correspond to the case that there are only scalar propagators on the loop. On
the contrary, the second and the third terms on the RHS of the third equation
contribute to loops with at least one gluon propagator.

The one-loop YMS integrands (in both single- and multi-trace cases) with
quadratic propagators, coming from the forward limit of the tree-level
amplitudes whose particles $+$ and $-$ are in the same trace and adjacent to
each other, are directly obtained from (5.14) and the discussions in section
7. Now we show that the tree-level CHY half integrands
$I_{\,\text{L}}^{\,\text{tree}}(+,Q,-,R;...||\boldsymbol{G})$ with particles
$+$ and $-$ separated by the scalars belonging to the same trace, can be
rewritten as

$\displaystyle\sum\limits_{\begin{subarray}{c}Q\in\text{cyc}\,\boldsymbol{\sigma}_{i}\\\
R\in\text{cyc}\,\boldsymbol{\sigma}_{j}\end{subarray}}I_{\,\text{L}}^{\text{tree}}(+,Q,-,R;\boldsymbol{\sigma}_{1};...;\cancel{\boldsymbol{\sigma}_{i}};...;\cancel{\boldsymbol{\sigma}_{j}};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G})$
(B.2) $\displaystyle=$
$\displaystyle\sum\limits_{\text{cyc}\,\boldsymbol{\rho}}\,\sum\limits_{R^{\text{T}}\in\text{cyc}\,\boldsymbol{\sigma}^{\text{T}}_{j}}(-1)^{|\boldsymbol{\sigma}_{j}|}I_{\,\text{L}}^{\text{tree}}(+,Q_{1},\\{Q_{2},...\\}\shuffle
R^{\text{T}},-;\boldsymbol{\sigma}_{1};...;\cancel{\boldsymbol{\sigma}_{i}};...;\cancel{\boldsymbol{\sigma}_{j}};...;\boldsymbol{\sigma}_{m}||\boldsymbol{G})\,,$

where the ordered sets $\boldsymbol{\sigma}_{i}$ and $\boldsymbol{\rho}$ in
the above summations are defined by $\boldsymbol{\sigma}_{i}=Q_{1},Q_{2},...$,
and $\boldsymbol{\rho}=Q_{1},\\{Q_{2},...\\}\shuffle R^{\text{T}}$. The
$|\boldsymbol{\sigma}_{j}|$, $\boldsymbol{\sigma}^{\text{T}}_{j}$ denote the
number of elements and the transpose of the permutation
$\boldsymbol{\sigma}_{j}$, respectively. To prove eq. (B.2), we arrange the
tree-level PT factors as

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{cyc}\,\boldsymbol{\alpha}\\\
\text{cyc}\,\boldsymbol{\beta}\end{subarray}}{\text{PT}}(+,\boldsymbol{\alpha},-,\boldsymbol{\beta})$
$\displaystyle=\sum\limits_{\begin{subarray}{c}\text{cyc}\,\boldsymbol{\alpha}\\\
\text{cyc}\,\boldsymbol{\beta}^{\text{T}}\end{subarray}}(-1)^{|\boldsymbol{\beta}|}\,{\text{PT}}(+,\boldsymbol{\alpha}\shuffle\boldsymbol{\beta}^{\text{T}},-)$
(B.3)
$\displaystyle=\sum\limits_{\text{cyc}\,\boldsymbol{\rho}}\sum\limits_{\text{cyc}\,\boldsymbol{\beta}^{\text{T}}}(-1)^{|\boldsymbol{\beta}|}\,{\text{PT}}(+,\alpha_{1},\\{\alpha_{2},...,\alpha_{s}\\}\shuffle\boldsymbol{\beta}^{\text{T}},-)\,,$

where $\boldsymbol{\alpha}=\alpha_{1},\alpha_{2},...,\alpha_{s}$,
$\boldsymbol{\rho}=\alpha_{1},\\{\alpha_{2},...,\alpha_{s}\\}\shuffle\boldsymbol{\beta}^{\text{T}}$.
The first equality is the KK relation, while the second can be checked by
comparing the permutations on both sides of this equation. For any permutation
on the LHS, one can always find it on the RHS of the second equality. Both
sides of this equation have the number of permutations
${(s+t)!\over(s-1)!(t-1)!}$, where $s$ and $t$ are the number of elements of
$\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$. For example, we choose
$\boldsymbol{\alpha}=1,2$ and $\boldsymbol{\beta}=3,4$, then both the LHS of
the first line and the RHS of the second line have the following $24$
permutations: 1234, 2341, 3412, 4123, 1324, 3241, 2413, 4132, 1342, 3421,
4213, 2134, 1243, 2431, 4312, 3124, 1423, 4231, 2314, 3142, 1432, 4321, 3214,
2143. This observation can be straightforwardly extended to general
permutations, therefore eq. (B.2) is proven.

This relation of PT factors can be considered as the CHY version of the one-
loop KK relation (the one for YM amplitudes was proposed in DelDuca:1999rs )

$\displaystyle
A^{\text{1-loop}}(\boldsymbol{\alpha};\boldsymbol{\beta})=\sum\limits_{\boldsymbol{\rho}\in\text{COP}(\boldsymbol{\alpha}\,\cup\,\boldsymbol{\beta}^{\text{T}})}(-1)^{|\boldsymbol{\beta}|}\,A^{\text{1-loop}}(\boldsymbol{\rho})\,,$
(B.4)

where $A^{\text{1-loop}}(\boldsymbol{\alpha};\boldsymbol{\beta})$ refers to a
one-loop multi-trace amplitude and $A^{\text{1-loop}}(\boldsymbol{\rho})$
denote a one-loop amplitude with one less trace. The
$\text{COP}(\boldsymbol{\alpha}\,\cup\,\boldsymbol{\beta}^{\text{T}})$ denotes
the summation of all possible permutations that keep $\boldsymbol{\alpha}$,
$\boldsymbol{\beta}^{\text{T}}$ in their own cyclic order.

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# Gravitational polarization of test-mass potential in equilibrium polytropic
sheets with non-negative polytropic indexes

Yuta Ito<EMAIL_ADDRESS>
###### Abstract

Gravitational polarization is examined for equilibrium self-gravitating
polytropic sheets perturbed by gravitational field due to test mass sheet. We
find equilibrium solutions to the corresponding perturbed Lane-Emden equations
for non-negative polytropic indexes. It is shown that gravitational
polarization may be observed even in a _finite_ extent of self-gravitating
systems in addition to previously discussed infinite systems. In the
polytropic sheets, the maximum gravitational amplification gets greater with a
higher polytropic index while the height at which the maximum amplification
occurs gets lower. The ratio of height change to the original height increases
with polytropic index. The last result constrains the linear approximation
method used for the present perturbation method.

###### keywords:

Polytropic sheet, Gravitational polarization, Collisional self-gravitating
systems

##  1 Introduction

Gravitational polarization of fields induced by test mass is an important but
little-explored fundamental concept in statistical dynamics of self-
gravitating systems. Polarization effects are generally known for dielectrics
under external electric fields and for electrolytes or plasmas perturbed by a
test-charge potential. In the systems, positive and negative charge
distributions are reconfigured so that applied electric fields are weaken or
even shielded. On the one hand, if test mass is added in a self-gravitating
system, gravitational field due to test mass could be reinforced (Miller,
1966; Gilbert, 1970). The mechanism is simple; gravitational field due to test
mass attracts ambient masses toward the test mass, the reconfigured mass
distribution then can strengthen the field. Gravitational polarization could
help understand the fundamental response of self-gravitating system to
external perturbations. Examples are a deformation of polarized halo due to
disc growth Murali and Tremaine (1998); Moody and Kalnajs (1999),
perturbations in massive haloes (Murali, 1999), stellar accretion and heavy
central astrophysical objects (Young, 1980; Murali and Tremaine, 1998; Quinlan
et al., 1995), interaction between “dressed” particles/stars (Heyvaerts, 2010;
Chavanis, 2012), and flyby of heavy objects (Vesperini and Weinberg, 2000).
These settings are realistic but complicated, so they are not suitable to
assess _pure_ effects and properties of gravitational polarization. The
simplest setting is to examine gravitation amplification is to add or move
test mass to the center of gravity in a self-gravitating system so that the
spatial symmetry is held through test-mass perturbation (Gilbert, 1970;
Goodman and Binney, 1984).

Discussing gravitational polarization of point-mass potential, however, is not
a straightforward topic. First, self-gravitating systems are inhomogeneous and
finite in size because of self-gravity. We can not resort to an infinite
homogeneous approximation, which often allows us to use simple mathematical
deductions. Such an approximation was applied to collisionless stellar systems
(Marochnik, 1968) and cold dark matters (Padmanabhan and Vasanthi, 1985) with
the Maxwellian distribution function for stars and particles. The works
suggested that gravitational potential due to test point mass behaves like
$\cos(k_{\text{J}}r)/r$, where $k_{\text{J}}$ is the Jeans wave number, and
$r$ the distance from test mass. The configuration, however, suffers from the
problem of “Jeans swindle” (e.g., Falco et al., 2013) which is an inconsistent
method unless we consider a cosmological setting. Second, particles and stars
orbit obeying self-consistent Newtonian mean-field potentials in self-
gravitating systems. This feature is especially important for stars in stellar
systems, such as galaxies, globular clusters, and nuclear star clusters. Stars
move around forming _smooth_ orbits that are not disturbed irregularly by
other stars on dynamical time scales (e.g., Binney and Tremaine, 2011). This
implies that we need to know explicitly analytical expressions to describe
orbits, such as orbital periods and isolating integrals. It is possible to
obtain the expressions only for limited cases. Goodman and Binney (1984) used
a harmonic oscillator as an approximation of stellar cluster core. A
successful discussion of gravitational amplification was made for a
collisionless isochrone (Gilbert, 1968) where an emerging test-point mass is
placed at the center of the isochrone holding the conservation of total mass.
Gilbert showed that gravitational field due to test mass is strengthened with
radius at small radii and reaches its maximum once. At larger radii, the
effect of amplification disappears as the density approaches zero. Third, even
with extensive numerical methods, the orbital effects can be handled only for
weak perturbation. To overcome the problem of orbital effects above, it is
possible to use Kalnajs (1977)’s matrix method for realistic self-gravitating
systems, such as the King models. However, the method applies only to weak
perturbation problems. Even for such limited setting, the method needs
exhaustive series expansions with sophisticated numerical schemes (Murali,
1999). It appears that no further works are found with this method for
gravitational polarization.

For this situation, we recently discussed gravitational polarization in
infinite collisional gaseous systems, namely the isothermal sheets, cylinder,
and sphere (Ito, 2023). There are advantages in examining the models. (i) We
can avoid the orbital effects of particles because of high collisionality in
the systems. (ii) As discussed in (Murali and Tremaine, 1998), gaseous (fluid)
system could show similar response features to collisionless systems. (iii)
Not only linear but non-linear analyses are easily executed because of simple
mathematical structures of the Lane-Emden equation. In (Ito, 2023), we found
in the isothermal sheets and cylinder that potentials due to test masses show
the same qualitative characteristics as that in collisionless isochrone. On
the one hand, the isothermal sphere showed an oscillatory gravitational field
due to test point mass similar to that in infinite homogeneous collisionless
systems with the Maxwellian distribution. The isothermal systems seems _nice_
models to discuss gravitational polarization compared to the previous results.
Hence, it would be reasonable to explore other collisional self-gravitating
systems as well for further discussions.

All the previous works focusing on gravitational polarization examined only
_infinite_ self-gravitating systems. Such systems are unrealistic in nature.
It is important to examine whether gravitational polarization may be observed
even in _finite_ systems as well. The present paper examines gravitational
polarization in equilibrium self-gravitating polytropic sheets. The mass
density and height of the sheet models are finite in the stratified direction
for polytropic index of $0\leq n<\infty$ (e.g., Horedt, 2000). The models are
important to understand stratification and fragmentation of self-gravitating
gaseous systems. Also, they are stable against radial perturbation. The last
feature may be less attractive for a statistical-dynamics point of view (Campa
et al., 2009) since the sheet models do not show exotic collective features,
such as negative specific heat. Yet, we believe that excluding those features
can make easier our understanding of gravitational polarization.

The goal of the present work is to show gravitational polarization in
polytropic sheets. We assume that test mass sheet is placed perpendicularly to
the stratified direction at the center of the sheets. We numerically solve a
linearlized Lane-Emden equation perturbed by potential due to test mass. To
account the finiteness of system size, we employ the method used for a tidal
effect on polytropic spheres (Chandrasekhar, 1933a, b). Our numerical results
show that the maximum of gravitational field gets greater with a higher
polytropic index in polytropic sheets. In the limit $n>>1$, the maximum
approaches that of the isothermal polytrope. We also show that the shortening
of polytropic sheets is more significant for higher polytropic indexes. This
result provides the limit of the linearlization approximation that we use for
test-mass perturbation.

The present paper is organized as follows. Section 2 explains polytropic sheet
models and the perturbation method to examine gravitational polarization.
Section 3 shows analytical results for polytropes of $n=0$ and $n=1$ to which
our perturbation method is applied. It also introduces measures of
gravitational amplification and shortening of polytropic sheets. Section 4
shows numerical results for $0\leq n<\infty$. Section 5 is Conclusion.

##  2 Polytropic sheets and its perturbation due to test mass

The present section first explains equilibrium polytropic sheets and a
perturbation method for the sheets due to test mass sheet. It then explains
the effect of shortening on the sheets.

###  2.1 Deriving the Lane-Emden equation for polytropic sheets

Polytropic sheet models are three-dimensional self-gravitating equilibrium
model composed of particles interacting via the pair-wise Newtonian potential
(e.g., Horedt, 2000). The self-consistent mean-field potential
$\Phi(\boldsymbol{r})$ is determined by mass density $\rho(\boldsymbol{r})$
via the Poisson equation

$\frac{\partial}{\partial\boldsymbol{r}}\cdot\left(\frac{\partial\Phi}{\partial\boldsymbol{r}}\right)=4\pi
G\rho(\boldsymbol{r}),$ (2.1)

where $G$ is the gravitational constant. In the equilibrium model, the
hydrostatic equation must hold;

$\frac{\partial
p}{\partial\boldsymbol{r}}+\rho(\boldsymbol{r})\frac{\partial\Phi}{\partial\boldsymbol{r}}=0,$
(2.2)

where $p(\boldsymbol{r})$ is the isotropic scalar pressure. We assume
polytropic sheets to satisfy the polytropic relation

$p=K\rho^{1+1/n},$ (2.3)

where $K$ is the polytropic constant and $n$ the polytropic index. Equations
(2.1), (2.2), and (2.3) are a set of closed equations for variables
$\rho(\boldsymbol{r})$, $p(\boldsymbol{r})$, and $\Phi(\boldsymbol{r})$.

With dimensionless variables

$\displaystyle\phi$ $\displaystyle\equiv$
$\displaystyle\frac{\Phi(\boldsymbol{r})}{K(n+1)\,\rho_{\text{c}}^{1/n}},$
(2.4) $\displaystyle\boldsymbol{\xi}$ $\displaystyle\equiv$
$\displaystyle\frac{\boldsymbol{r}}{L_{\text{c}}}\equiv\left(\frac{4\pi
G}{K(n+1)}\rho_{\text{c}}^{1-1/n}\right)^{1/2}\boldsymbol{r},$ (2.5)

where $\rho_{\text{c}}$ is the central density of $\rho(r)$, equations (2.1),
(2.2), and (2.3) reduce to the Lane-Emden equation for polytropes of index $n$

$\frac{\partial}{\partial\boldsymbol{\xi}}\cdot\left(\frac{\partial\phi}{\partial\boldsymbol{\xi}}\right)-\left(-\phi\right)^{n}=0,$
(2.6)

where the following relations are used

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\rho_{\text{c}}\left(-\phi\right)^{n},$ (2.7) $\displaystyle p$
$\displaystyle=$ $\displaystyle
K\rho_{\text{c}}^{1+1/n}\left(-\phi\right)^{n+1}.$ (2.8)

The polytropic sheets are stratified in a direction, say the $z$-direction.
Equation (2.6) for stratified sheets along the $z$-axis is written as

$\frac{\text{d}^{2}\phi}{\text{d}\,z^{2}}-\left(-\phi\right)^{n}=0.$ (2.9)

The boundary conditions (BCs) for equation (2.9) are

$\phi(z=0)=-1,\qquad\phi^{\prime}(z=0)=0.$ (2.10)

Our interest is polytrope sheets of $0\leq n<\infty$ whose potential
monotonically decreases with height and reaches zero at a finite height. We
label the maximum height, or the first zero of the potential, as
$z_{\text{M}}$ hereafter. Their mass density also has the same characteristics
as the potential except for $n=0$; the density is uniform in the polytrope of
$n=0$. Figure 1 shows mass density distributions at $z>0$ for selected
polytropic indexes.

$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$0$$0.2$$0.4$$0.6$$0.8$$1$$z$$\rho$$n=1$$n=2$$n=5$$n=20$$n=100$
Figure 1: Mass density distribution of the polytropic sheet models.

###  2.2 Perturbation method for the polytropic sheets

The present mathematical formulation for gravitational amplification is the
same as that for gravitational tides (Chandrasekhar, 1933b) except for the
location of test mass. The _bare_ gravitational field strength due to test
mass sheet is constant. Hence, even if it is placed outside a polytrope, it
does not cause tidal effect. On the one hand, if it is placed at the center of
gravity of the polytrope, it causes gravitational amplification.

We use the perturbation method developed for the isothermal systems (Ito,
2023). Imagine that test mass sheet is added at $z=0$ on the $xy$-plane in a
polytropic sheet model. Equation (2.9) is modified as

$\frac{\text{d}^{2}\phi}{\text{d}\,z^{2}}-\left(-\phi\right)^{n}=\epsilon\delta(z),$
(2.11)

where $\epsilon$ is a small parameter defined as

$\epsilon\equiv\dfrac{m_{\text{p}}}{\rho_{\text{c}}L_{\text{c}}},$ (2.12)

where $m_{\text{p}}$ is test mass. Expand equation (2.11) with $\epsilon$ as

$\frac{\text{d}^{2}}{\text{d}\,z^{2}}\left(\phi_{\text{o}}+\epsilon\delta\phi\right)-\left(-\phi_{\text{o}}-\epsilon\delta\phi\right)^{n}=\epsilon\delta(z),$
(2.13)

where $\phi_{\text{o}}$ is the unperturbed potential of polytropic sheet model
and $\epsilon\delta\phi$ the potential deviated from $\phi_{\text{o}}$ due to
test mass. At the order of $\epsilon$, we have a linearized Lane-Emden
equation for $\delta\phi$;

$\frac{\text{d}^{2}\delta\phi}{\text{d}\,z^{2}}+n\left(-\phi_{\text{o}}\right)^{n-1}\delta\phi=\delta(z).$
(2.14)

We introduce the following potential

$W(z)\equiv\delta\phi(z)-\frac{1}{2}|z|,$ (2.15)

so that equation (2.11) does not include the delta function

$\frac{\text{d}^{2}W}{\text{d}\,z^{2}}+n\left(-\phi_{\text{o}}\right)^{n-1}\left(W+\frac{1}{2}|z|\right)=0.$
(2.16)

With equation (2.15), there is no gravitational field due to $W(z)$ at $z=0$.
Hence, the BC is

$W^{\prime}(z=0)=0.$ (2.17)

Another BC is determined so that the effective mass $m^{*}$, the partial sum
of reconfigured sheet masses, is zero at $|z|=z_{\text{M}}$. The effective
mass for the polytropic sheet is at $z$

$\displaystyle m^{*}(z)$ $\displaystyle=$
$\displaystyle-2\int_{0}^{z}n\left(-\phi_{\text{o}}\left(z^{\prime}\right)\right)^{n-1}\left(W\left(z^{\prime}\right)+\frac{1}{2}|z^{\prime}|\right)\text{d}z^{\prime},$
(2.18) $\displaystyle=$ $\displaystyle
2\left(W^{\prime}(z)-W^{\prime}(0)\right).$

The mass $m^{*}(z)$ must be zero at $z=z_{\text{M}}$ since test mass
$m_{\text{p}}$ is not included in the reconfigured masses. Accordingly, the BC
is at $|z|=z_{\text{M}}$

$W^{\prime}(|z|=z_{\text{M}})=0.$ (2.19)

Our fundamental numerical strategy is to find potential $W(z)$ by solving
equation (2.16) with two BCs (2.17) and (2.19) after finding the unperturbed
potential $\phi_{\text{o}}$ by solving equation (2.9) with BCs in equation
(2.10).

###  2.3 Polytropic sheet shortening and surface boundary condition

Gravitational potential due to test mass sheet pulls polytropic sheets in the
$z$-direction toward the test mass, which results in a shortening of sheet
thickness. The discussion of polytrope shortening is essentially the same as
that of distorted polytrope due to tidal effects (Chandrasekhar, 1933b). We
first introduce the new maximum height of polytropic sheets due to
perturbation

$z_{\text{s}}\equiv z_{\text{M}}+\epsilon\delta z,$ (2.20)

where $\delta z$ is the deviation from $z_{\text{M}}$ at order of $\epsilon$
and expected to take a negative value. Expand the new equilibrium potential

$\phi(z)=\phi_{\text{o}}(z)+\epsilon\delta\phi(z).$ (2.21)

around at $z_{\text{M}}$ using equation (2.20) as follows

$\phi(z_{\text{s}})=\phi_{\text{o}}(z_{\text{M}})+\epsilon\left(\phi_{\text{o}}^{\prime}(z_{\text{M}})\,\delta
z+\delta\phi(z_{\text{M}})\right)+\mathcal{O}\left(\epsilon^{2}\right).$
(2.22)

The new equilibrium potential $\phi(z)$ must be zero at $z_{\text{s}}$. Since
$\phi_{\text{o}}(z_{\text{M}})$ is zero, we have the condition

$\delta
z=-\frac{\delta\phi(z_{\text{M}})}{\phi_{\text{o}}^{\prime}(z_{\text{M}})}=-\frac{W(z_{\text{M}})+\frac{1}{2}z_{\text{M}}}{\phi_{\text{o}}^{\prime}(z_{\text{M}})}.$
(2.23)

Using equation (2.20), we next expand the derivative of potential $\phi(z)$

$\phi^{\prime}(z)=\phi_{\text{o}}^{\prime}(z)+\epsilon\delta\phi^{\prime}(z)$
(2.24)

to the order of $\epsilon$ as follows

$\phi^{\prime}(z_{\text{s}})=\phi_{\text{o}}^{\prime}(z_{\text{M}})+\epsilon\left(\phi_{\text{o}}^{\prime\prime}(z_{\text{M}})\,\delta
z+\delta\phi^{\prime}(z_{\text{M}})\right).$ (2.25)

With function $W(z)$ in equation (2.15), another condition is given on the
surface of the perturbed polytrope as

$\phi_{\text{o}}^{\prime\prime}(z_{\text{M}})\,\delta
z+W^{\prime}(z_{\text{M}})=0.$ (2.26)

With equation (2.23), we hence obtain the boundary condition on the surface

$-\frac{W(z_{\text{M}})+\frac{1}{2}z_{\text{M}}}{\phi_{\text{o}}^{\prime}(z_{\text{M}})}\,\phi_{\text{o}}^{\prime\prime}(z_{\text{M}})+W^{\prime}(z_{\text{M}})=0.$
(2.27)

Equation (2.27) is a generalized BC of that in equation (2.19). For polytropes
of $1<n<\infty$, the term $\phi_{\text{o}}^{\prime\prime}(z_{\text{M}})$ is
zero because of equation (2.9). This means that the boundary condition on a
shortened polytrope surface is not affected by the shortening, or $\delta z$.
The effect of shortening should appear in distorted mass distribution when
$\phi_{\text{o}}^{\prime\prime}(z_{\text{M}})$ is non-zero. Such an example is
that polytropic sheets are inserted between two pressurized mediums. Another
example is that mass density does not reach zero at $z_{\text{M}}$, which
corresponds with the polytropic sheet of $n=0$ in the present work.

##  3 Gravitational polarization in the polytropic sheet models of $n=0$ and
$n=1$

The present section explains gravitational amplification of polytropic sheets
of $n=0$ and $n=1$. We provide the explicitly analytical form of the
amplification for the models using the perturbation method explained in
Section 2. We also introduce measures of gravitational amplification and
shortening of polytropic sheets.

###  3.1 Gravitational polarization for $n=0$

The Lane-Emden equation for polytrope of $n=0$ is

$\frac{\text{d}^{2}\phi}{\text{d}\,z^{2}}-1=0.$ (3.1)

With the BCs in equation (2.10), the solution reads

$\phi_{\text{o}}(z)=-1+\frac{z^{2}}{2},$ (3.2)

and the maximum height is

$z_{\text{M}}=\sqrt{2}.$ (3.3)

The equation for the perturbed polytrope of $n=0$ is the Laplace equation

$W^{\prime\prime}(z)=0,$ (3.4)

with the BCs (2.17) and (2.27). The solution reads

$W(z)=-\frac{1}{\sqrt{2}}.$ (3.5)

This means that gravitational field due to test mass sheet is not amplified
and that only the reference value of potential constant is determined as
$-1/\sqrt{2}$. Also, no shortening of the polytropic sheets occurs;

$\delta z=0.$ (3.6)

With the above analysis, we confirm the consistency of the BC (2.27). If the
BC (2.19) is used instead, $W(z)$ is undetermined.

###  3.2 Gravitational polarization for $n=1$

The Lane-Emden equation for polytropic sheet of $n=1$ is

$\frac{\text{d}^{2}\phi}{\text{d}\,z^{2}}+\phi=0.$ (3.7)

With the BCs in equation (2.10), the solution reads

$\phi_{\text{o}}(z)=-\cos z.$ (3.8)

From equation (2.16), the linearized Lane-Emden equation for potential $W(z)$
is

$\frac{\text{d}^{2}W}{\text{d}\,z^{2}}+\left(W+\frac{1}{2}|z|\right)=0.$ (3.9)

With two BCs (2.17) and (2.19), the solution is on $z>0$

$W(z)=\frac{1}{\sqrt{2}}\sin\left(z-\frac{\pi}{4}\right)-\frac{1}{2}z.$ (3.10)

In analogy with the standard electromagnetism, we introduce the following
gravitational “susceptibility”

$\chi(z)\equiv-2\delta\phi^{\prime}(z).$ (3.11)

The quantity $\chi(z)$ is the ratio of the amplified to the _bare_
gravitational field due to test mass. If $\chi(z)$ is greater than one, it
means that gravitational field is amplified. It appears suitable as a measure
of amplification, we hence call $\chi(z)$ the _gravitational amplification_ of
fields due to test mass hereafter. For $n=1$, we have

$\chi(z)=\sqrt{2}\cos\left(z-\frac{\pi}{4}\right).$ (3.12)

The height $z_{\chi.\text{max}}$ at which gravitational amplification reaches
the maximum is $\pi/4$. The maximum of gravitational amplification,
$\chi_{\text{max}}$, is $\sqrt{2}$. We also introduce the average of $\chi(z)$

$\chi_{\text{ave}}\equiv\frac{1}{2z_{\text{M}}}\int^{z_{\text{M}}}_{-z_{\text{M}}}\chi\left(z^{\prime}\right)\,\text{d}z^{\prime}=\frac{\delta\phi(z_{\text{M}})-\delta\phi(0)}{z_{\text{M}}}.$
(3.13)

For $n=1$, $z_{\text{M}}$ is $\pi/2$. So, the value of $\chi_{\text{ave}}$ is

$\chi_{\text{ave}}=\frac{4}{\pi}.$ (3.14)

We next introduce the following quantity to examine the degree of shortening
of polytropic sheets

$\eta\equiv\frac{\delta
z}{z_{\text{M}}}=-\frac{W(z_{\text{M}})+\frac{1}{2}z_{\text{M}}}{z_{\text{M}}\,\phi_{\text{o}}^{\prime}(z_{\text{M}})}.$
(3.15)

For $n=1$, the ratio reads

$\eta=-\frac{1}{2\sqrt{2}}.$ (3.16)

The above analytical results are important. First, they are used to confirm
our numerical calculation. Second, unlike other indexes, the perturbed
polytrope of $n=1$ does not explicitly depend on the unperturbed density as
seen in equation (3.9). Polytropes with $n=1$ may give a hint of _pure_
gravitational-polarization effect. Comparing our result to the collisionless
homogeneous systems (Marochnik, 1968; Padmanabhan and Vasanthi, 1985), it
seems that “pure” gravitational polarization causes a sinusoidal
amplification.

##  4 Numerical results

The present section shows numerical results for perturbed polytropic sheets
due to test mass sheet. We first explain the local characteristics of
gravitational amplification and then global ones in the sheets. We lastly
assess the shrinkage of sheet thickness in the $z$-direction.

###  4.1 Maximum gravitational amplification and its location

Figure 2 depicts numerical values of gravitational amplification in selected
polytropic sheets perturbed by test mass. The characteristics of $\chi(z)$ are
alike among the polytropic sheets between $n=0.01$ and $n=500$. The
amplification is maximized only once along the $z$-direction and weak near
$z=0$ and $z=z_{\text{M}}$. This feature is qualitatively the same as
previously reported gravitational amplification in collisionless isochrone
(Gilbert, 1970) and isothermal cylinder and sheet (Ito, 2023).

In the polytropic sheets, the maximum amplification $\chi_{\text{max}}$
increases with index $n$ and asymptotically reaches $1.68$ (Figure 3). This
value approximately equals the gravitational amplification obtained for the
isothermal sheet model. The isothermal sheet may be considered the case of
$n\to\infty$ in the polytropic sheets (e.g., Horedt, 2000). On the one hand,
the location of the maximum amplification gets closer to $z=0$ as $n$
increases (Figure 4). This would reflect a feature of unperturbed polytropic
sheets that the mass distribution is concentrated more near $z=0$ as $n$
increases (Figure 1).

$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$1$$1.2$$1.4$$1.6$$z$$\chi$$n=1$$n=2$$n=5$$n=20$$n=100$
Figure 2: Gravitational amplification $\chi$ due to test mass sheet in the
polytropic sheet models.
$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$1$$1.2$$1.4$$1.6$$1.8$$n$$\chi_{\text{max}}$
Figure 3: Maximum gravitational amplification $\chi_{\text{max}}$ against
polytropic index $n$ for the polytropic sheet model.
$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$0$$0.5$$1$$n$$r_{\chi.\text{max}}$
Figure 4: Location of $\chi_{\text{max}}$ against polytropic index $n$ for the
polytropic sheet models.

###  4.2 Average gravitational amplification

The numerical value of $\chi_{\text{ave}}$ is shown in Figure 5 against
polytropic index $n$. As easily expected, the polytrope of $n=1$ takes a
relatively high value of $\chi_{\text{ave}}$ since it does not depend on its
unperturbed mass distribution and system size. The maximum of
$\chi_{\text{ave}}$ occurs when $n\simeq 1.6$. This would be the outcome of
the two effects; a moderately high index $n$ and slowly decaying density
$\rho_{\text{o}}(z)$ in the perturbed density $(\propto
n(-\phi_{\text{o}}(z))^{n-1}=n\rho_{\text{o}}^{1-1/n}(z))$ in equation (2.14).
For this, we first assume from the result of Section 3.1 that if the perturbed
density is close to zero then the amplification is not effective. For example,
for very low $n(\ll 1)$, mass density distribution slowly diverges at large
distances, but the perturbed density is still low at every height $z$ due to
the low $n$. For higher $n(\gg 1)$, $\chi(z)$ can reach large values, but the
average is low because of large polytrope height and rapid decay in the
unperturbed density. We hence may expect the perturbed density to be more
effective for amplification when $n$ is the order of 1.

$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$1$$1.1$$1.2$$1.3$$n$$\chi_{\text{ave}}$
Figure 5: Average gravitational amplification $\chi_{\text{ave}}$ against
polytropic index $n$ for the polytropic sheet models.

###  4.3 Shortening of the polytropic sheets

The _bare_ gravitational field due to test mass is constant $(=1/2)$
throughout polytropic sheets. The higher sheets can be deviated more largely
from the original to the new equilibrium position because of low pressure.
Polytropes with higher polytropic index hence experience more significant
shortening in the $z$-direction (Figure 6). The ratio $\eta$ is well
approximated by $\sim n^{0.492}$ for high indexes. This means that we need a
very small value of $\epsilon$ for a high polytropic index. For example, to
achieve 1$\%$ of shrinkage, or $\epsilon\delta z/z_{\text{M}}=0.01$, the
necessary value of $\epsilon$ is $\sim 0.01n^{-0.492}$; $\epsilon\sim 10^{-5}$
for $n=500$. It is obvious that the linear approximation of potential
$\phi(z)$ may break down for high $n$. This break-down could be reasonable by
seeing the expanded form of $\phi(z)$ in equation (2.21). The perturbation
potential $\epsilon\delta\phi(z)$ increases like $\epsilon z/2$ as $z$
increases while $\phi_{\text{o}}(z)$ decreases. The linearization would
successfully apply to polytropes with shorter heights. To hold the linear
approximation for any $n$, polytropes must embedded in a pressurized medium.

$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{-1}$$10^{0}$$10^{1}$$n$$-\eta$$0.39n^{0.492}$
Figure 6: Negative of ratio $\eta$ of $\delta z$ to $z_{\text{M}}$ against
polytropic index $n$ for the polytropic sheet models. A guideline is depicted
together as an approximation of $\eta$ for high index $n$.

##  5 Conclusion

The present work discussed gravitational polarization of fields induced by
test mass sheet instantaneously placed at $z=0$ in polytropic sheets of $0\leq
n<\infty$. It showed that gravitational amplification occurs even in a
_finite_ extent of equilibrium self-gravitating systems in addition to
previously reported infinite collisional and collisionless systems. We first
obtained analytical results for polytropic sheets of $n=0$ and $n=1$. The
former model is a special case in which neither system shrinkage nor
gravitational amplification does occur. The latter model is the simplest model
that possesses basic features of gravitational amplification. It also helps
confirm our numerical results.

Our numerical results showed that gravitational amplification $\eta$ in
polytropic sheets are qualitatively the same as those in collisionless
isochrone, isothermal sheet, and isothermal cylinder. The amplification is one
at $z=0$, gets greater with height, and reaches its maximum value. After
passing the maximum, it then decreases with height and equals one at the
maximum height. We found in the polytropic sheets that the maximum of $\eta$
increases with polytropics index $n$ and approaches that of the isothermal
sheet model. The height at which the maximum occurs gets shorter for a higher
polytropic index. We also found that the average of gravitational
amplification is maximized around at $n=1.6$ and it gets weaker as $n$ gets
greater or less. Lastly, we computed the shortening of the polytropic sheets.
The ratio of the height change to maximum height becomes greater with index
$n$. It is approximately proportional to $n$ when $n\gg 1$. This analysis
provided a constraint on the present linear approximation.

In the future paper, we will extend the present work to nonlinear cases. We
may also apply the present method to polytropic cylinder and sphere.
Especially, the latter model is unstable to radial perturbation. We will
confine the isothermal sphere in a rigid wall or by a pressurized medium or
polytrope. The present formulation, especially for $n=0$, can be readily
arranged for such confined systems. Another interesting extension work would
be to move test mass from the original equilibrium polytrope to the center of
gravity rather than adding extra test mass. Such a process was already
discussed in Gilbert (1970) for collisionless isochrone. The discussion can be
useful to understand the discreteness of self-gravitating systems; it will
help understand how the “statistical term (Gilbert, 1968)” affects
gravitational fields due to test mass and how the fields disappear near the
surface of self-gravitating systems.

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