#  $N$-photon amplitudes in a plane-wave background

James P. Edwards  Christian Schubert

###### Abstract

We use the worldline formalism to derive master formulas for the one-loop $N$
\- photon amplitudes in a plane-wave background, for both scalar and spinor
QED. This generalises previous work by Ilderton and Torgrimsson for the vacuum
polarisation case, although with some change in methodology since, instead of
evaluating the path integral on the semi-classical trajectory, we use the
special kinematics of the plane-wave background to uncover the crypto-gaussian
character of this type of worldline path integral.

[label1]organization=Instituto de Física y Matemáticas,Universidad Michoacana
de San Nicolás de Hidalgo, addressline=Edificio C-3, Apdo. Postal 2-82,
city=Morelia, postcode=58040, state=Michoacán, country=Mexico

##  1 Introduction

In strong-field QED, there are two external field configurations that play a
special rôle: the constant field, and the plane-wave one. Both are not only
important from the physics point of view, but also special mathematically,
since they allow for an exact solution of the Dirac equation in the field,
which makes it possible to perform non-perturbative calculations in such
fields (see, e.g., [1, 2]). Nevertheless, beyond the simplest special cases
such calculations tend to be extremely lengthy and tedious [3, 4, 5, 6, 7, 8,
9]. For example, the one-loop QED vertex in a plane-wave field has been
calculated only very recently [10].

For the constant-field case there exists an alternative approach, based on
Feynman’s worldline path integral formulation of QED [11, 12] and concepts
originally borrowed from string theory [13, 14], that has been shown to offer
various technical advantages for closed-loop photonic processes [15, 16, 17,
18] and recently also for amplitudes involving open scalar [19] and fermion
[20] lines – for reviews of this formalism see [21, 22].

The plane-wave case has attracted much attention in recent years because of
its relevance for laser physics [23, 24, 25]. However, the application of the
worldline formalism to this case has turned out to be less straightforward. A
calculation of the scalar and spinor QED vacuum polarisations along these
lines was achieved by A. Ilderton and G. Torgrimsson [26], but it is not
obvious how to extend their approach to the general $N$-photon amplitudes.
Here we will use a slightly different approach, based on a direct rewriting of
the worldline path integral as a gaussian one, to construct compact master
formulas for the scalar and spinor QED $N$-photon amplitudes in a plane-wave
background.

We start in the following section with a short summary of the worldline
representation of the $N$-photon amplitudes in vacuum (for details see [21]).
The following two sections are devoted to the derivation of master formulas
for the $N$-photon amplitudes in a plane-wave background, first for scalar and
then for spinor QED. As a check, in section 5 we work out the $N=2$ cases and
recover the results of [26]. In the final section we shortly summarize our
results and point out possible generalisations.

##  2 $N$-photon amplitudes in the worldline formalism

The starting point for the calculation of the scalar QED $N$-photon amplitudes
in the worldline formalism is Feynman’s [11] worldline representation of the
one-loop effective action $\Gamma_{\rm scal}[A]$:

$\displaystyle\Gamma_{\rm scal}[A]$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dT}{T}\ \,{\rm
e}^{-m^{2}T}\int_{x(T)=x(0)}Dx\,\,{\rm
e}^{-\int_{0}^{T}d\tau\bigl{(}\frac{\dot{x}^{2}}{4}+ie\,\dot{x}\cdot
A(x)\bigr{)}}$ (1)

The path integral runs over all closed trajectories in spacetime obeying the
periodicity condition $x(T)=x(0)$ in proper time. The $N$-photon amplitudes
are obtained from this by expanding the “interaction exponential” and Fourier
transformation, which leads to the “vertex operator representation” of the
$N$-photon amplitude:

$\displaystyle\Gamma_{\rm
scal}(k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N})$ $\displaystyle=$
$\displaystyle(-ie)^{N}\int_{0}^{\infty}\frac{dT}{T}\,\,{\rm
e}^{-m^{2}T}\int_{x(0)=x(T)}Dx\,\,{\rm
e}^{-\int_{0}^{T}d\tau\frac{\dot{x}^{2}}{4}}$ (2) $\displaystyle\times
V^{\gamma}_{\rm scal}[k_{1},\varepsilon_{1}]\cdots V^{\gamma}_{\rm
scal}[k_{N},\varepsilon_{N}]$

Here each photon is represented by the following photon vertex operator,
integrated along the trajectory:

$\displaystyle V^{\gamma}_{\rm scal}[k,\varepsilon]$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{T}d\tau\,\varepsilon\cdot\dot{x}(\tau)\,{\,{\rm
e}}^{ik\cdot x(\tau)}$ (3)

After a formal exponentiation $\varepsilon_{i}\cdot\dot{x}_{i}\,\,{\rm
e}^{ik_{i}\cdot x_{i}}=\,{\rm e}^{\varepsilon_{i}\cdot\dot{x}_{i}+ik_{i}\cdot
x_{i}}\mid_{\varepsilon_{i}}$ the path integral can be done by gaussian
integration using the basic correlator

$\displaystyle\langle
x^{\mu}(\tau)x^{\nu}(\tau^{\prime})\rangle=-G(\tau,\tau^{\prime})\delta^{\mu\nu},\quad
G(\tau,\tau^{\prime})\equiv\mid\tau-\tau^{\prime}\mid-\frac{(\tau-\tau^{\prime})^{2}}{T}\,.$
(4)

This results in the following “Bern-Kosower representation” of the $N$-photon
amplitude [27, 13, 14],

$\displaystyle\Gamma_{\rm
scal}(k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N})$ $\displaystyle=$
$\displaystyle{(-ie)}^{N}{(2\pi)}^{D}\delta(\sum
k_{i}){\displaystyle\int_{0}^{\infty}}\frac{dT}{T}{(4\pi
T)}^{-\frac{D}{2}}e^{-m^{2}T}$ (5)
$\displaystyle\times\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\exp\biggl{\\{}\sum_{i,j=1}^{N}\Bigl{[}\frac{1}{2}G_{ij}k_{i}\cdot
k_{j}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\Bigr{]}\biggr{\\}}\Bigl{|}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{N}}$

Here a ‘dot’ denotes a derivative acting on the first variable,

$\displaystyle\dot{G}(\tau,\tau^{\prime})$ $\displaystyle=$ $\displaystyle{\rm
sign}(\tau-\tau^{\prime})-2\frac{(\tau-\tau^{\prime})}{T},\quad\ddot{G}(\tau,\tau^{\prime})=2{\delta}(\tau-\tau^{\prime})-\frac{2}{T}$
(6)

and we abbreviate $G_{ij}\equiv G(\tau_{i},\tau_{j})$ etc. The factor ${(4\pi
T)}^{-\frac{D}{2}}$ represents the free Gaussian path integral determinant
factor, and the ${(2\pi)}^{D}\delta(\sum k_{i})$ factor is produced by the
integration over the zero mode
$x_{0}^{\mu}\equiv\frac{1}{T}\int_{0}^{T}d\tau\,x^{\mu}(\tau)$ of the path
integral. The exponential must still be expanded and only the terms be
retained that contain each polarisation vector $\varepsilon_{i}$ linearly:

$\displaystyle\exp\bigl{\\{}\cdot\bigr{\\}}\bigl{|}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{N}}\quad\equiv{(-i)}^{N}P_{N}(\dot{G}_{ij},\ddot{G}_{ij})\exp\biggl{[}\frac{1}{2}\sum_{i,j=1}^{N}G_{ij}k_{i}\cdot
k_{j}\biggr{]}$ (7)

with certain polynomials $P_{N}$.

For spinor QED, a generalization of (1) suitable for analytical calculations
is given by the Feynman-Fradkin representation [12, 28]

$\displaystyle\Gamma_{\rm spin}[A]$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{\displaystyle\int_{0}^{\infty}}\frac{dT}{T}\int
Dx\int
D\psi\,e^{-\int_{0}^{T}d\tau\bigl{(}\frac{1}{4}\dot{q}^{2}+\frac{1}{2}\psi\cdot\dot{\psi}+\frac{1}{2}ie\,\dot{x}\cdot
A(x)-ie\psi^{\mu}F_{{\mu\nu}}(x)\psi^{\nu}\bigr{)}}\,.$

Here $\psi^{\mu}(\tau)$ is a Lorentz vector whose components are Grassmann
functions, $\\{\psi^{\mu}(\tau),\psi^{\nu}(\tau^{\prime})\\}=0$, and the path
integral $\int D\psi$ has to be taken over antiperiodic such functions,
$\psi^{\mu}(T)=-\psi^{\mu}(0)$. Note that it is already gaussian as it stands.

Applying the same procedure as for the scalar case above, one obtains the
following generalization of (2) to the spinor QED case:

$\displaystyle\Gamma_{\rm
spin}(k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(-ie)^{N}\int_{0}^{\infty}\frac{dT}{T}\,\,{\rm
e}^{-m^{2}T}\int_{x(0)=x(T)}Dx\,e^{-\int_{0}^{T}d\tau\frac{\dot{x}^{2}}{4}}$
(9) $\displaystyle\times\int_{A}D\psi\,\,{\rm
e}^{-\int_{0}^{T}d\tau\frac{1}{2}\psi\cdot\dot{\psi}}V^{\gamma}_{\rm
spin}[k_{1},\varepsilon_{1}]\cdots V^{\gamma}_{\rm
spin}[k_{N},\varepsilon_{N}]\,.$

The photon vertex operator for spinor QED $V^{\gamma}_{\rm spin}$ differs from
the scalar one (3) by a second term representing the interaction of the
fermion spin with the photon,

$\displaystyle V^{\gamma}_{\rm spin}[k,\varepsilon]$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{T}d\tau\,\Bigl{[}\varepsilon\cdot\dot{x}(\tau)-i\psi(\tau)\cdot
f\cdot\psi(\tau)\Bigr{]}\,\,{\rm e}^{ik\cdot x(\tau)}$ (10)

with $f^{\mu\nu}\equiv k^{\mu}\varepsilon^{\nu}-\varepsilon^{\mu}k^{\nu}$ the
photon field-strength tensor. Thus the $N$-photon amplitude is naturally
obtained in terms of a spin-orbit decomposition

$\displaystyle\Gamma_{{\rm
spin},N}=\sum_{S=0}^{N}\Gamma_{NS}\,,\quad\Gamma_{NS}=\sum_{\\{i_{1}i_{2}\ldots
i_{S}\\}}\Gamma_{NS}^{\\{i_{1}i_{2}\ldots i_{S}\\}}\,,$ (11)

where $S$ denotes the number of spin interactions, and the sum
$\sum_{\\{i_{1}i_{2}\ldots i_{S}\\}}$ runs over all choices of $S$ out of the
$N$ photons as the ones assigned to those interactions. It is then
straightforward to arrive at the following master formula for
$\Gamma_{NS}^{\\{i_{1}i_{2}\ldots i_{S}\\}}$:

$\displaystyle\Gamma_{NS}^{\\{i_{1}i_{2}\ldots i_{S}\\}}$
$\displaystyle=-2(-e)^{N}\int_{0}^{\infty}\frac{dT}{T}\,{(4\pi T)}^{-{D\over
2}}\,{\rm e}^{-m^{2}T}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\,$
$\displaystyle\times
W(k_{i_{1}},\varepsilon_{i_{1}};\ldots;k_{i_{S}},\varepsilon_{i_{S}})P_{NS}^{\\{i_{1}i_{2}\ldots
i_{S}\\}}{\rm e}^{\frac{1}{2}\sum_{i,j=1}^{N}G_{ij}k_{i}\cdot k_{j}}\,.$ (12)

Here the polynomials $P_{NS}^{\\{i_{1}i_{2}\ldots i_{S}\\}}$ are now defined
by (compare with (7))

$\displaystyle{\rm
e}^{\sum_{i,j=1}^{N}\big{(}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\big{)}}\big{|}_{\varepsilon_{i_{1}}=\cdots=\varepsilon_{i_{S}}=0}\big{|}_{\varepsilon_{i_{S+1}}\cdots\varepsilon_{i_{N}}}\equiv(-i)^{N-S}P_{NS}^{\\{i_{1}i_{2}\ldots
i_{S}\\}}\,,$ (13)

where the notation on the left-hand side means that one first sets the
polarisation vectors $\varepsilon_{i_{1}},\ldots,\varepsilon_{i_{S}}$ equal to
zero, and then selects all the terms linear in the surviving polarisation
vectors. In particular, one has the extremal cases
$P^{\\{\\}}_{N0}=P_{N}\,,P^{\\{12\ldots N\\}}_{NN}=1$. For the Wick-
contraction of the spin interaction terms we have introduced the notation

$\displaystyle
W(k_{i_{1}},\varepsilon_{i_{1}};\ldots;k_{i_{S}},\varepsilon_{i_{S}})\equiv\Bigl{\langle}\psi_{i_{1}}\cdot
f_{i_{1}}\cdot\psi_{i_{1}}\cdots\psi_{i_{S}}\cdot
f_{i_{S}}\cdot\psi_{i_{S}}\Bigr{\rangle}\quad\quad$ (14)

to be evaluated with the basic correlator

$\displaystyle\langle\psi^{\mu}(\tau)\psi^{\nu}(\tau^{\prime})\rangle=\frac{1}{2}G_{F}(\tau,\tau^{\prime})\delta^{\mu\nu},\quad
G_{F}(\tau,\tau^{\prime})\equiv{\rm sgn}(\tau-\tau^{\prime})\,.$ (15)

This object possesses the following closed-form description. Define a “Lorentz
cycle of length $n$” $Z_{n}$ by

$\displaystyle Z_{n}(i_{1}i_{2}\ldots i_{n})$ $\displaystyle\equiv$
$\displaystyle\Bigl{(}\frac{1}{2}\Bigr{)}^{\delta_{n2}}{\rm
tr}\Bigl{(}\prod_{j=1}^{n}f_{i_{j}}\Bigr{)}$ (16)

and a “fermionic bi-cycle of length $n$” by

$\displaystyle G_{F}(i_{1}i_{2}\ldots i_{n})\equiv
G_{Fi_{1}i_{2}}G_{Fi_{2}i_{3}}\cdots G_{Fi_{n}i_{1}}Z_{n}(i_{1}i_{2}\ldots
i_{n})\qquad(n\geq 2)\,.$ (17)

Then we can write

$\displaystyle W(k_{1},\varepsilon_{1};\ldots;k_{S},\varepsilon_{S})$
$\displaystyle=$ $\displaystyle\sum_{\rm
partitions}(-1)^{cy}G_{F}(i_{1}i_{2}\ldots i_{n_{1}})G_{F}(i_{n_{1}+1}\ldots
i_{n_{1}+n_{2}})\cdots$ (18) $\displaystyle\times
G_{F}(i_{n_{1}+\ldots+n_{cy-1}+1}\ldots i_{n_{1}+\ldots+n_{cy}})\,.$

Here the sum runs over all inequivalent possibilities to distribute the
indices $1,\ldots,S$ among the arguments of any number $cy$ of bi-cycles, and
$n_{k}$ denotes the length of the bi-cycle $k$. Working out (18) up to $S=4$,
we find

$\displaystyle W(k_{1},\varepsilon_{1})$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle W(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2})$
$\displaystyle=$ $\displaystyle-G_{F}(12)\,,$ $\displaystyle
W(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2};k_{3},\varepsilon_{3})$
$\displaystyle=$ $\displaystyle-G_{F}(123)\,,$ $\displaystyle
W(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2};k_{3},\varepsilon_{3};k_{4},\varepsilon_{4})$
$\displaystyle=$ $\displaystyle-G_{F}(1234)-G_{F}(1243)-G_{F}(1324)$
$\displaystyle+G_{F}(12)G_{F}(34)+G_{F}(13)G_{F}(24)+G_{F}(14)G_{F}(23)\,.$

##  3 $N$-photon amplitude in a plane-wave background (scalar QED)

In general, a plane-wave field can be defined by a vector potential $A(x)$ of
the form

$\displaystyle eA_{\mu}(x)=a_{\mu}(n\cdot x)$ (20)

where $n^{\mu}$ is a null vector,

$\displaystyle n^{2}=0$ (21)

and, as is usual, we will further impose the light-front gauge condition

$\displaystyle n\cdot a=0$ (22)

Note that we absorb the charge $e$ in the definition of $a_{\mu}$. Repeating
the procedure of the previous section with the addition of the potential
$a_{\mu}$ to the worldline Lagrangian, we straightforwardly get a
representation of the $N$-photon amplitude in the plane-wave background that
generalizes the vacuum formula (2),

$\displaystyle\Gamma_{\rm scal}(\\{{k_{i},\varepsilon_{i}\\}};a)$
$\displaystyle=$ $\displaystyle(-ie)^{N}\int_{0}^{\infty}\frac{dT}{T}\,\,{\rm
e}^{-m^{2}T}\int Dx\,\,{\rm
e}^{-\int_{0}^{T}d\tau\bigl{[}\frac{\dot{x}^{2}}{4}+i\dot{x}^{\mu}a_{\mu}(n\cdot
x)\bigr{]}}$ (23) $\displaystyle\times V^{\gamma}_{\rm
scal}[k_{1},\varepsilon_{1}]\cdots V^{\gamma}_{\rm
scal}[k_{N},\varepsilon_{N}]$

Fixing the zero-mode problem as usual by separating off the average position
$x_{0}^{\mu}$ of the trajectory, $x^{\mu}(\tau)=x_{0}^{\mu}+q^{\mu}(\tau)$, we
note that, differently from the vacuum case, it now appears not only in the
exponents of the vertex operators, but also in the argument of $a_{\mu}(n\cdot
x)$. Thus it will now be convenient to introduce (euclidean) light-cone
coordinates adapted to the null vector $n^{\mu}$. Thus we set
$n^{\mu}\equiv\frac{1}{\sqrt{2}}(0,0,1,i)$, and define $x^{+}\equiv n\cdot
x=\frac{1}{\sqrt{2}}(x^{3}+ix^{4})$ (“light-front time”) and
$x^{-}\equiv\frac{1}{\sqrt{2}}(-x^{3}+ix^{4})$. We will further denote
$x^{\perp}\equiv(x^{1},x^{2})$. Defining also
$k^{\pm}\equiv\frac{1}{\sqrt{2}}(\pm k^{3}+ik^{4})$, and using the
decomposition

$\displaystyle k\cdot x=-k^{+}x^{-}-k^{-}x^{+}+k^{1}x^{1}+k^{2}x^{2}$ (24)

allows us to integrate out $x_{0}^{\mu}$ but for its $x_{0}^{+}$ component:

$\displaystyle\int Dx\,\,{\rm
e}^{-\int_{0}^{T}d\tau\bigl{[}\frac{\dot{x}^{2}}{4}+i\dot{x}^{\mu}a_{\mu}(n\cdot
x)\bigr{]}}V^{\gamma}_{\rm scal}[k_{1},\varepsilon_{1}]\cdots V^{\gamma}_{\rm
scal}[k_{N},\varepsilon_{N}]$
$\displaystyle=(2\pi)^{3}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{1}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{2}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{+}\bigr{)}\int_{-\infty}^{\infty}dx_{0}^{+}\,{\rm
e}^{-ix_{0}^{+}\sum_{i=1}^{N}k_{i}^{-}}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}$
$\displaystyle\times\int Dq\,{\rm
e}^{-\int_{0}^{T}d\tau\bigl{[}\frac{\dot{q}^{2}}{4}+i\dot{q}^{\mu}a_{\mu}(x_{0}^{+}+n\cdot
q(\tau))\bigr{]}}\,\,{\rm e}^{\sum_{i=1}^{N}(ik_{i}\cdot
q_{i}+\varepsilon_{i}\cdot\dot{q}_{i})}\Big{|}_{\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{N}}\,.$
(25)

The calculation of the functional integral at first sight looks like an
intractable problem, since the integration variable $q(\tau)$ appears in the
argument of the unknown function $a_{\mu}$. In [26] this problem was solved
for the two-point case using the fact that the plane-wave path integral
possesses the gaussian property that its semiclassical approximation is exact.
For the $N$-point generalization, we find it more convenient to exhibit the
crypto-gaussian nature of the path integral using the relations (21) and (22).
In principle, we could do the functional integral by expanding

$\displaystyle\,{\rm e}^{-\int_{0}^{T}d\tau i\dot{q}\cdot a(x_{0}^{+}+n\cdot
q(\tau))}$ $\displaystyle=$ $\displaystyle 1-i\int_{0}^{T}d\tau\,\dot{q}\cdot
a(x_{0}^{+}+n\cdot q(\tau))$ (26)
$\displaystyle+\frac{(-i)^{2}}{2!}\int_{0}^{T}d\tau\,\dot{q}_{1}\cdot
a(x_{0}^{+}+n\cdot q_{1})\int_{0}^{T}d\tau^{\prime}\,\dot{q}_{2}\cdot
a(x_{0}^{+}+n\cdot q_{2})+\cdots$

and then Taylor-expanding

$\displaystyle a_{\mu}(x_{0}^{+}+n\cdot
q_{m})=a_{\mu}(x_{0}^{+})+a^{\prime}_{\mu}(x_{0}^{+})(n\cdot
q_{m})+\frac{1}{2!}a^{\prime\prime}_{\mu}(x_{0}^{+})(n\cdot q_{m})^{2}+\cdots$
(27)

(Note that we use a ‘prime’ for the derivative of a function with respect to
its argument, while the ‘dot’ will be used for the total derivative with
respect to proper time.) Now, we observe that a factor $n\cdot q_{m}$ can
neither be Wick contracted with another such factor because of (21), nor with
a factor of $\dot{q}_{n}\cdot a^{(k)}(x_{0}^{+})$ because of (22). Thus each
$n\cdot q_{m}$ has to be contracted with the exponential $\,{\rm
e}^{\sum_{i=1}^{N}(ik_{i}\cdot q_{i}+\varepsilon_{i}\cdot\dot{q}_{i})}$, and
this will convert it into

$\displaystyle n\cdot q_{m}\longrightarrow
n\cdot\sum_{i=1}^{N}\bigl{[}-iG_{mi}k_{i}+\dot{G}_{mi}\varepsilon_{i}\bigr{]}$
(28)

We can then resum (27) into

$\displaystyle a_{\mu}(x_{0}^{+}+n\cdot q_{m})\longrightarrow
a_{\mu}\Bigl{(}x_{0}^{+}+n\cdot\sum_{i=1}^{N}[-iG_{mi}k_{i}+\dot{G}_{mi}\varepsilon_{i}]\Bigr{)}$
(29)

and subsequently also (26),

$\displaystyle\,{\rm e}^{-\int_{0}^{T}d\tau i\dot{q}\cdot a(x_{0}^{+}+n\cdot
q(\tau))}\longrightarrow\,{\rm e}^{-\int_{0}^{T}d\tau i\dot{q}\cdot
a\bigl{(}x_{0}^{+}+n\cdot\sum_{i=1}^{N}[-iG(\tau,\tau_{i})k_{i}+\dot{G}(\tau,\tau_{i})\varepsilon_{i}]\bigr{)}}$

where we can now, with some abuse of notation, replace

$\displaystyle
a_{\mu}\Bigl{(}x_{0}^{+}+n\cdot\sum_{i=1}^{N}[-iG(\tau,\tau_{i})k_{i}+\dot{G}(\tau,\tau_{i})\varepsilon_{i}]\Bigr{)}\equiv
a_{\mu}(\tau)$ (31)

Thus we have removed the functional integration variable from the argument of
$a_{\mu}$, and converted the functional integral (25) into a gaussian one. Now
the usual “completing-the-square” procedure can be applied, and yields

$\displaystyle\int Dq\,{\rm
e}^{-\int_{0}^{T}d\tau\bigl{[}\frac{\dot{q}^{2}}{4}+i\dot{q}\cdot
a(\tau)\bigr{]}}\,\,{\rm e}^{\sum_{i=1}^{N}(ik_{i}\cdot
q_{i}+\varepsilon_{i}\cdot\dot{q}_{i})}$ $\displaystyle={(4\pi T)}^{-{D\over
2}}\,{\rm
e}^{-\frac{1}{2}\int_{0}^{T}d\tau\int_{0}^{T}d\tau^{\prime}\ddot{G}(\tau,\tau^{\prime})a(\tau)\cdot
a(\tau^{\prime})-\sum_{i=1}^{N}\int_{0}^{T}d\tau\bigl{[}\dot{G}(\tau,\tau_{i})a(\tau)\cdot
k_{i}+i\ddot{G}(\tau,\tau_{i})a(\tau)\cdot\varepsilon_{i}\bigr{]}}$
$\displaystyle\quad\times\,{\rm
e}^{\sum_{i,j=1}^{N}\bigl{[}\frac{1}{2}G_{ij}k_{i}\cdot
k_{j}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\bigr{]}}$
(32)

The first term in the exponent on the right-hand-side can, introducing the
worldline average

$\displaystyle\langle\langle f\rangle\rangle\equiv\frac{1}{T}\int_{0}^{T}d\tau
f(\tau)$ (33)

and using (6) be rewritten as

$\displaystyle\frac{1}{2}\int_{0}^{T}d\tau\int_{0}^{T}d\tau^{\prime}\ddot{G}(\tau,\tau^{\prime})a(\tau)\cdot
a(\tau^{\prime})=T\Bigl{(}\langle\langle a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\Bigr{)}$ (34)

Similarly, we can rewrite

$\displaystyle\sum_{i=1}^{N}\int_{0}^{T}d\tau\,\ddot{G}(\tau,\tau_{i})a(\tau)\cdot\varepsilon_{i}=2\sum_{i=1}^{N}\Bigl{(}a(\tau_{i})-\langle\langle
a\rangle\rangle\Bigr{)}\cdot\varepsilon_{i}$ (35)

For the integral involving $\dot{G}(\tau,\tau_{i})$, we introduce the periodic
integral function

$\displaystyle
I_{\mu}(\tau)\equiv\int_{0}^{\tau}d\tau^{\prime}\Bigl{(}a_{\mu}(\tau^{\prime})-\langle\langle
a_{\mu}\rangle\rangle\Bigr{)}$ (36)

Integrating by parts, we get

$\displaystyle\sum_{i=1}^{N}\int_{0}^{T}d\tau\,\dot{G}(\tau,\tau_{i})a(\tau)\cdot
k_{i}=-2\sum_{i=1}^{N}k_{i}\cdot\bigl{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\bigr{)}$ (37)

Putting the pieces together, we get the following master formula for the
scalar QED $N$-photon amplitude in a plane-wave background111The reader
familiar with the worldline formalism may wonder why we could drop the
additive constant $\frac{T}{6}$ from this Green’s function, which is customary
but relies on momentum conservation. It is easy to verify that here, in light-
cone coordinates, the removal of the constant requires only that
$\sum_{i=1}^{N}k_{i}^{+,\perp}=0$, not full momentum conservation.

$\displaystyle\Gamma_{\rm scal}(\\{{k_{i},\varepsilon_{i}\\}};a)$
$\displaystyle=$
$\displaystyle(-ie)^{N}(2\pi)^{3}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{1}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{2}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{+}\bigr{)}\int_{-\infty}^{\infty}dx_{0}^{+}\,{\rm
e}^{-ix_{0}^{+}\sum_{i=1}^{N}k_{i}^{-}}$
$\displaystyle\times\int_{0}^{\infty}\frac{dT}{T}\,{(4\pi T)}^{-{D\over
2}}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\,{\rm
e}^{\sum_{i,j=1}^{N}\bigl{[}\frac{1}{2}G_{ij}k_{i}\cdot
k_{j}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\bigr{]}}$
$\displaystyle\times\,{\rm e}^{-\bigl{(}m^{2}+\langle\langle
a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\bigr{)}T+2\sum_{i=1}^{N}k_{i}\cdot\bigl{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\bigr{)}-2i\sum_{i=1}^{N}\bigl{(}a(\tau_{i})-\langle\langle
a\rangle\rangle\bigr{)}\cdot\varepsilon_{i}}\Bigl{|}_{\varepsilon_{1}\cdots\varepsilon_{N}}$
(38)

Note that the appearance of the polarization vectors in the argument of
$a_{\mu}$ makes it still messy to extract the terms linear in all of them.
Further substantive simplification can be achieved by choosing the
$\varepsilon_{i}$ such as to obey

$\displaystyle n\cdot\varepsilon_{i}=0\quad(i=1,\ldots,N)$ (39)

which is possible for generic momenta by a gauge transformation, and will be
assumed for the rest of this paper. This will reduce (31) to

$\displaystyle
a_{\mu}(\tau)=a_{\mu}\Bigl{(}x_{0}^{+}-i\sum_{i=1}^{N}G(\tau,\tau_{i})k_{i}^{+}\Bigr{)}$
(40)

The master formula (38) can then be written more explicitly as

$\displaystyle\Gamma_{\rm scal}(\\{{k_{i},\varepsilon_{i}\\}};a)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!(-e)^{N}(2\pi)^{3}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{1}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{2}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{+}\bigr{)}\\!\\!\int_{-\infty}^{\infty}dx_{0}^{+}\,{\rm
e}^{-ix_{0}^{+}\sum_{i=1}^{N}k_{i}^{-}}$ (41)
$\displaystyle\times\int_{0}^{\infty}\frac{dT}{T}\,{(4\pi T)}^{-{D\over
2}}\,{\rm e}^{-\bigl{(}m^{2}+\langle\langle a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\bigr{)}T}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\,$
$\displaystyle\times\mathfrak{P}_{N}\,{\rm
e}^{\frac{1}{2}\sum_{i,j=1}^{N}G_{ij}k_{i}\cdot
k_{j}+2\sum_{i=1}^{N}k_{i}\cdot\bigl{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\bigr{)}}$

where the polynomials $\mathfrak{P}_{N}$ are defined by (compare (7))

$\displaystyle{\rm
e}^{\sum_{i,j=1}^{N}\big{(}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\big{)}-2i\sum_{i=1}^{N}\bigl{(}a(\tau_{i})-\langle\langle
a\rangle\rangle\bigr{)}\cdot\varepsilon_{i}}\big{|}_{\varepsilon_{1}\cdots\varepsilon_{N}}\equiv(-i)^{N}{\mathfrak{P}}_{N}\,.$
(42)

##  4 $N$-photon amplitude in a plane-wave background (spinor QED)

Proceeding to the spinor QED case, the same argument that we applied above to
$eA_{\mu}=a_{\mu}$ can be used to convert also the argument of the
$eF_{\mu\nu}=n_{\mu}a^{\prime}_{\nu}-a^{\prime}_{\mu}n_{\nu}$ appearing in the
spin part of the worldline Lagrangian in (LABEL:spinorQEDgrass) in the same
way as in (29). Thus the only new element is that the fermionic Wick-
contraction rule (14) now has to be calculated with a generalised worldline
Green’s function inverting the field-dependent operator

$\displaystyle{\cal O}\equiv\frac{\delta_{\mu\nu}}{2}{d\over
d\tau}+ia^{\prime}_{\mu}(\tau)n_{\nu}-in_{\mu}a^{\prime}_{\nu}(\tau)$ (43)

The appropriate generalisation of (15) is

$\displaystyle\langle\psi^{\mu}(\tau)\psi^{\nu}(\tau^{\prime})\rangle=\frac{1}{2}\mathfrak{G}_{F}^{\mu\nu}(\tau,\tau^{\prime}),$
(44)

where

$\mathfrak{G}_{F}^{\mu\nu}(\tau,\tau^{\prime})\equiv\biggl{\\{}\delta^{\mu\nu}+2in^{\mu}{\cal
J}^{\nu}(\tau,\tau^{\prime})+2i{\cal
J}^{\mu}(\tau^{\prime},\tau)n^{\nu}+2\Bigl{[}{\cal
J}^{2}(\tau,\tau^{\prime})-\frac{T^{2}}{4}\langle\langle
a^{\prime}\rangle\rangle^{2}\Bigr{]}n^{\mu}n^{\nu}\biggr{\\}}G_{F}(\tau,\tau^{\prime})$
(45)

and we have further defined

$\displaystyle J_{\mu}(\tau)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\tau}d\tau^{\prime}\Bigl{(}a^{\prime}_{\mu}(\tau^{\prime})-\langle\langle
a^{\prime}_{\mu}\rangle\rangle\Bigr{)}\,,$ (46) $\displaystyle{\cal
J}_{\mu}(\tau,\tau^{\prime})$ $\displaystyle\equiv$ $\displaystyle
J_{\mu}(\tau)-J_{\mu}(\tau^{\prime})-\frac{T}{2}\dot{G}(\tau,\tau^{\prime})\langle\langle
a^{\prime}_{\mu}\rangle\rangle\,.$ (47)

Note that the modified Green’s function has a non-zero coincidence limit,

$\mathfrak{G}_{F}^{\mu\nu}(\tau,\tau)=-iT\Big{(}n^{\mu}\langle\langle
a^{\prime\nu}\rangle\rangle-\langle\langle a^{\prime\mu}\rangle\rangle
n^{\nu}\Big{)}$ (48)

and satisfies the anti-symmetry relation
$\mathfrak{G}_{F}(\tau^{\prime},\tau)=-\mathfrak{G}_{F}^{\intercal}(\tau,\tau^{\prime})$.
Proceeding as in the vacuum case, we get a spin-orbit decomposition as in (11)
with

$\displaystyle\Gamma_{NS}^{\\{i_{1}i_{2}\ldots i_{S}\\}}$ $\displaystyle=$
$\displaystyle-2(-e)^{N}(2\pi)^{3}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{1}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{2}\bigr{)}\delta\bigl{(}\sum_{i=1}^{N}k_{i}^{+}\bigr{)}\int_{-\infty}^{\infty}dx_{0}^{+}\,{\rm
e}^{-ix_{0}^{+}\sum_{i=1}^{N}k_{i}^{-}}$ (49)
$\displaystyle\times\int_{0}^{\infty}\frac{dT}{T}\,{(4\pi T)}^{-{D\over
2}}\,{\rm e}^{-\bigl{(}m^{2}+\langle\langle a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\bigr{)}T}\prod_{i=1}^{N}\int_{0}^{T}d\tau_{i}\,$
$\displaystyle\times\mathfrak{W}(k_{i_{1}},\varepsilon_{i_{1}};\ldots;k_{i_{S}},\varepsilon_{i_{S}})\mathfrak{P}_{NS}^{\\{i_{1}i_{2}\ldots
i_{S}\\}}{\rm e}^{\frac{1}{2}\sum_{i,j=1}^{N}G_{ij}k_{i}\cdot
k_{j}+2\sum_{i=1}^{N}k_{i}\cdot\bigl{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\bigr{)}}\,.$

The polynomials $\mathfrak{P}_{NS}$ now are defined by

$\displaystyle{\rm
e}^{\sum_{i,j=1}^{N}\big{(}-i\dot{G}_{ij}\varepsilon_{i}\cdot
k_{j}+\frac{1}{2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\big{)}-2i\sum_{i=1}^{N}\bigl{(}a(\tau_{i})-\langle\langle
a\rangle\rangle\bigr{)}\cdot\varepsilon_{i}}\big{|}_{\varepsilon_{i_{1}}=\cdots=\varepsilon_{i_{S}}=0}\big{|}_{\varepsilon_{i_{S+1}}\cdots\varepsilon_{i_{N}}}\equiv(-i)^{N-S}{\mathfrak{P}}_{NS}^{\\{i_{1}i_{2}\ldots
i_{S}\\}}\,,$ (50)

and
$\mathfrak{W}(k_{i_{1}},\varepsilon_{i_{1}};\ldots;k_{i_{S}},\varepsilon_{i_{S}})$
denotes the correlator (14) evaluated with the modified fermionic Wick
contraction (44). For the calculation of this correlator we can still use the
cycle decomposition formula (18), only that the fermionic bicycle (17), now
must be replaced by

$\displaystyle\mathfrak{G}_{F}(i_{1}i_{2}\dots i_{n})$
$\displaystyle~{}\equiv~{}\bigl{(}\frac{1}{2}\bigr{)}^{\delta_{n2}}\textrm{tr}(f_{i_{1}}\cdot{\mathfrak{G}}_{Fi_{1}i_{2}}\cdot
f_{i_{2}}\cdot{\mathfrak{G}}_{Fi_{2}i_{3}}\cdots
f_{i_{n}}\cdot{\mathfrak{G}}_{Fi_{n}i_{1}})~{}~{}$ (51)

Note that, differently from the case of a constant external field [16, 18],
the fermionic path-integral determinant factor is not affected by the presence
of the plane-wave field, and remains at its free value $2^{\frac{D}{2}}$.

##  5 The case $N=2$

As a check, let us show that the above master formulas correctly reproduce the
results of [26] for the $N=2$ case. We first give the general off-shell
results before specialising to the on-shell helicity flip process studied
there. We work throughout with the gauge choice (39) for convenience and note
that momentum conservation in the $+$ direction gives $k_{2}^{+}=-k_{1}^{+}$.

###  5.1 Off-shell

Using the notation introduced in (11), for $\Gamma_{20}$ we find from (50)

$\displaystyle\mathfrak{P}_{20}^{\\{\\}}$
$\displaystyle=\dot{G}_{12}\dot{G}_{21}\varepsilon_{1}\cdot
k_{2}\varepsilon_{2}\cdot
k_{1}-\ddot{G}_{12}\varepsilon_{1}\cdot\varepsilon_{2}$
$\displaystyle+2\Big{(}\dot{G}_{12}\varepsilon_{1}\cdot
k_{2}\varepsilon_{2}\cdot\big{(}a(\tau_{2})-\langle\langle
a\rangle\rangle\big{)}+\dot{G}_{21}\varepsilon_{1}\cdot\big{(}a(\tau_{1})-\langle\langle
a\rangle\rangle\big{)}\varepsilon_{2}\cdot k_{1}\Big{)}$
$\displaystyle+4\varepsilon_{1}\cdot\big{(}a(\tau_{1})-\langle\langle
a\rangle\rangle\big{)}\varepsilon_{2}\cdot\big{(}a(\tau_{2})-\langle\langle
a\rangle\rangle\big{)}$ (52) $\displaystyle\mathfrak{W}()$
$\displaystyle=1\,,$ (53)

which is sufficient to produce the scalar QED result when substituted into
(12). Furthermore, for the spinor case we also require

$\displaystyle\mathfrak{P}_{21}^{\\{1\\}}$
$\displaystyle=\dot{G}_{21}\varepsilon_{2}\cdot
k_{1}+2\varepsilon_{2}\cdot\big{(}a(\tau_{2})-\langle\langle
a\rangle\rangle\big{)}$ (54)
$\displaystyle\mathfrak{W}(k_{1},\varepsilon_{1})$
$\displaystyle=-\frac{1}{2}\textrm{tr}\big{(}\mathfrak{G}_{F}(\tau_{1},\tau_{1})\cdot
f_{1}\big{)}=-iTk_{1}^{+}\varepsilon_{1}\cdot\langle\langle
a^{\prime}\rangle\rangle$ (55)

which, with the corresponding results under $1\leftrightarrow 2$, can be used
to construct $\Gamma_{21}$. Finally, we determine $\Gamma_{22}$ from

$\displaystyle\mathfrak{P}_{22}^{\\{12\\}}$ $\displaystyle=1$ (56)
$\displaystyle\mathfrak{W}(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2})$
$\displaystyle=-\mathfrak{G}_{F}(12)+\frac{1}{4}\textrm{tr}\big{(}\mathfrak{G}_{F}(\tau_{1},\tau_{1})\cdot
f_{1}\big{)}\textrm{tr}\big{(}\mathfrak{G}_{F}(\tau_{2},\tau_{2})\cdot
f_{2}\big{)}\,$ (57)

where ($\mathcal{J}_{ij}\equiv\mathcal{J}(\tau_{i},\tau_{j})$)

$\displaystyle-\mathfrak{G}_{F}(12)$
$\displaystyle=2k_{1}^{+}k_{2}^{+}\Big{[}\varepsilon_{1}\cdot\varepsilon_{2}\Big{(}\mathcal{J}_{12}^{2}-\frac{T^{2}}{4}\langle\langle
a^{\prime}\rangle\rangle^{2}\Big{)}+2\varepsilon_{1}\cdot\mathcal{J}_{21}\varepsilon_{2}\cdot\mathcal{J}_{12}\Big{]}\sigma_{12}\sigma_{21}$
$\displaystyle-2ik_{1}^{+}\Big{[}\varepsilon_{1}\cdot
k_{2}\varepsilon_{2}\cdot\mathcal{J}_{12}-\varepsilon_{1}\cdot\varepsilon_{2}k_{2}\cdot\mathcal{J}_{12}\Big{]}\sigma_{12}\sigma_{21}$
$\displaystyle-2ik_{2}^{+}\Big{[}\varepsilon_{2}\cdot
k_{1}\varepsilon_{1}\cdot\mathcal{J}_{21}-\varepsilon_{1}\cdot\varepsilon_{2}k_{1}\cdot\mathcal{J}_{21}\Big{]}\sigma_{12}\sigma_{21}$
$\displaystyle-\frac{1}{2}\textrm{tr}\big{(}f_{1}\cdot
f_{2}\big{)}\sigma_{12}\sigma_{21}\,.$ (58)

These lead to a spin-orbit decomposition

$\displaystyle\Gamma_{\textrm{spin},2}(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2})$
$\displaystyle=-2e^{2}\int_{-\infty}^{\infty}dx_{0}^{+}\,\,{\rm
e}^{-ix_{0}^{+}(k_{1}+k_{2})^{-}}\int_{0}^{\infty}\frac{dT}{T}(4\pi
T)^{-\frac{D}{2}}$ $\displaystyle\times{\rm e}^{-\big{(}m^{2}+\langle\langle
a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\big{)}T}\int_{0}^{T}d\tau_{1}\int_{0}^{T}d\tau_{2}\,{\rm
e}^{G_{12}k_{1}\cdot
k_{2}+2\sum_{i=1}^{2}k_{i}\cdot\bigl{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\bigr{)}}$
$\displaystyle\times\Big{\\{}\mathfrak{P}_{20}^{\\{\\}}+\mathfrak{W}(k_{1},\varepsilon_{1})\mathfrak{P}_{21}^{\\{1\\}}+\mathfrak{W}(k_{2},\varepsilon_{2})\mathfrak{P}_{21}^{\\{2\\}}+\mathfrak{W}(k_{1},\varepsilon_{1};k_{2}\varepsilon_{2})\Big{\\}}\,,$
(59)

where we have omitted the momentum conserving $\delta$-functions in the $+$
and $\perp$ directions. The term in the exponent
$\sum_{i=1}^{2}k_{i}\cdot\big{(}I(\tau_{i})-\langle\langle
I\rangle\rangle\big{)}$ at the two-point level could be removed by imposing on
$a_{\mu}$, instead of (22), the stronger gauge condition of full
transversality. To see this, it is easiest to return to (37). Using the
conservation of momentum along the $+$ and transversal directions together
with (40), and choosing a function $b_{\mu}(x)$ such that
$b^{\prime}_{\mu}=a_{\mu}$, we have

$\displaystyle\sum_{i=1}^{2}\int_{0}^{T}d\tau\,\dot{G}(\tau,\tau_{i})a(\tau)\cdot
k_{i}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}d\tau\,(\dot{G}(\tau,\tau_{1})-\dot{G}(\tau,\tau_{2}))a(\tau)\cdot
k_{1}$ (60) $\displaystyle=$
$\displaystyle\frac{i}{k_{1}^{+}}\int_{0}^{T}d\tau\frac{d}{d\tau}k_{1}\cdot
b\Bigl{(}x_{0}^{+}-i\sum_{i=1}^{N}G(\tau,\tau_{i})k_{i}^{+}\Bigr{)}$
$\displaystyle=$ $\displaystyle\frac{i}{k_{1}^{+}}k_{1}\cdot(b(T)-b(0))=0$

###  5.2 On-shell

In the on-shell case and for $N=2$ photons we gain additional simplifications
due to the mass shell condition which, by conservation of momentum in the $+$
and $\perp$ directions, implies the additional condition
$k_{2}^{-}=-k_{1}^{-}$ so that $k_{1}=k=-k_{2}$ with $k^{2}=0$. This removes
the exponent $\,{\rm e}^{G_{12}k_{1}\cdot k_{2}}$ from (59). Further imposing
the transversality conditions $\varepsilon_{1}\cdot
k=0=k\cdot\varepsilon_{2}$, the components of the spin orbit decomposition
reduce to

$\displaystyle\mathfrak{P}_{20}^{\\{\\}}$
$\displaystyle\rightarrow-\ddot{G}_{12}\varepsilon_{1}\cdot\varepsilon_{2}+4\varepsilon_{1}\cdot\big{(}a_{1}-\langle\langle
a\rangle\rangle\big{)}\varepsilon_{2}\cdot\big{(}a_{2}-\langle\langle
a\rangle\rangle\big{)}$ (61) $\displaystyle\mathfrak{P}_{21}^{\\{1\\}}$
$\displaystyle\rightarrow 2\varepsilon_{2}\cdot\big{(}a_{2}-\langle\langle
a\rangle\rangle\big{)},\quad\mathfrak{P}_{21}^{\\{2\\}}\rightarrow
2\varepsilon_{1}\cdot\big{(}a_{1}-\langle\langle a\rangle\rangle\big{)}$ (62)
$\displaystyle\mathfrak{W}(k_{i},\varepsilon_{i})$ $\displaystyle\rightarrow-
iTk^{+}\varepsilon_{i}\cdot\langle\langle a\rangle\rangle$ (63)
$\displaystyle\mathfrak{W}(k_{1},\varepsilon_{1};k_{2}\varepsilon_{2})$
$\displaystyle\rightarrow
k^{+2}\Bigl{[}2\varepsilon_{1}\cdot\varepsilon_{2}\big{(}\mathcal{J}_{12}^{2}-\frac{T^{2}}{4}\langle\langle
a^{\prime}\rangle\rangle^{2}\big{)}+4\varepsilon_{1}\cdot\mathcal{J}_{12}\varepsilon_{2}\cdot\mathcal{J}_{21}+T^{2}\varepsilon_{1}\cdot\langle\langle
a^{\prime}\rangle\rangle\varepsilon_{2}\cdot\langle\langle
a^{\prime}\rangle\rangle\Big{]}$ (64)

###  5.3 Helicity flip

For the purpose of comparing with the helicity-flip calculation of [26] we can
further put $\varepsilon_{1}\cdot\varepsilon_{2}=0$, resulting in

$\displaystyle\Gamma_{\textrm{spin},2}(k_{1},\varepsilon_{1};k_{2},\varepsilon_{2})$
$\displaystyle=-2e^{2}\int_{-\infty}^{\infty}dx_{0}^{+}\int_{0}^{\infty}\frac{dT}{T}(4\pi
T)^{-\frac{D}{2}}$ $\displaystyle\times{\rm e}^{-\big{(}m^{2}+\langle\langle
a^{2}\rangle\rangle-\langle\langle
a\rangle\rangle^{2}\big{)}T}\Big{\\{}4\varepsilon_{1}\cdot\big{(}a_{1}-\langle\langle
a\rangle\rangle\big{)}\varepsilon_{2}\cdot\big{(}a_{2}-\langle\langle
a\rangle\rangle\big{)}$
$\displaystyle-2iTk^{+}\Big{[}\varepsilon_{1}\cdot\langle\langle
a\rangle\rangle\varepsilon_{2}\cdot\big{(}a_{2}-\langle\langle
a\rangle\rangle\big{)}-\varepsilon_{1}\cdot\big{(}a_{1}-\langle\langle
a\rangle\rangle\big{)}\varepsilon_{2}\cdot\langle\langle
a\rangle\rangle\Big{]}$
$\displaystyle+k^{+2}\Big{[}4\varepsilon_{1}\cdot\mathcal{J}_{12}\varepsilon_{2}\cdot\mathcal{J}_{21}+T^{2}\varepsilon_{1}\cdot\langle\langle
a^{\prime}\rangle\rangle\varepsilon_{2}\cdot\langle\langle
a^{\prime}\rangle\rangle\Big{]}\Big{\\}}\,.$ (65)

This indeed correctly reproduces the parameter integrals determined in [26] as
can be easily seen using the integral representations

$\displaystyle\mathcal{J}_{12\mu}$
$\displaystyle=-\frac{1}{2}\sigma_{12}\int_{0}^{T}d\tau\,\sigma(\tau-\tau_{1})\sigma(\tau-\tau_{2})a^{\prime}_{\mu}(\tau)$
$\displaystyle\mathcal{J}_{12}^{2}-\frac{T^{2}}{4}\langle\langle
a^{\prime}\rangle\rangle^{2}$
$\displaystyle=-\frac{1}{2}\sigma_{12}\int_{0}^{T}d\tau\int_{0}^{T}d\tau^{\prime}\,\sigma(\tau-\tau_{1})\sigma(\tau-\tau^{\prime})\sigma(\tau^{\prime}-\tau_{2})a^{\prime}(\tau)\cdot
a^{\prime}(\tau^{\prime})\,.$ (66)

##  6 Summary and Outlook

We have used the worldline formalism to construct a master formula for the
$N$-photon amplitudes in a general plane-wave background field, for both
scalar and spinor QED. We believe that it compares favourably with other
available techniques for calculations in plane-wave backgrounds. As usual in
applications of the worldline formalism to QED, it unifies the scalar and
spinor cases in the sense that any spinor QED calculation yields the
corresponding scalar QED quantity as a side result. In particular, with the
formalism developed here a calculation of the photon-photon scattering
amplitudes in a plane-wave background should be quite feasible. In a more
extensive publication, we will give a more detailed derivation, including an
alternative approach using the worldline super formalism for the spinor QED
case, and explore the $N=3$ and $N=4$ cases. It should also be interesting to
generalize the mapping of worldline averages to spacetime averages, introduced
in [26], to the $N$-point case. Also under consideration is the extension of
the formalism to the open-line case, i.e. the photon-dressed scalar and spinor
propagators in a plane-wave background.

Acknowledgements: We thank Anton Ilderton for helpful correspondence
concerning some of the details in [26].

## References

  * [1] E.S. Fradkin, D.M. Gitman, S.M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum, Springer 1991.
  * [2] W. Dittrich and H. Gies, Probing the Quantum Vacuum. Perturbative Effective Action Approach in Quantum Electrodynamics and its Application, Springer Tracts Mod. Phys. 166 (2000) 1.
  * [3] A. I. Nikishov and V. I. Ritus, Zh. Eksp. Teor. Fiz. 46 (1964) 776 [Sov. Phys. JETP 19 (1964) 529].
  * [4] A. I. Nikishov and V. I. Ritus, Zh. Eksp. Teor. Fiz. 46 (1964) 1768 [Sov. Phys. JETP 19 (1964) 1191].
  * [5] P. Baier and P. Breitenlohner, Nuov. Cim. 47 (1967) 261.
  * [6] S. L. Adler, Ann. Phys. (N.Y.) 67 (1971) 599.
  * [7] W.Y. Tsai, Phys. Rev. D 10 (1974) 1342.
  * [8] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Zh. Eksp. Teor. Fiz. 67 (1974) 453 [Sov. Phys. JETP 40 (1975) 225].
  * [9] V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Zh. Eksp. Teor. Fiz. 68 (1975) 405 [Sov. Phys. JETP 41 (1975) 198].
  * [10] A. Di Piazza and M. A. Lopez-Lopez, Phys. Rev. D 102 (2020) 116012, arXiv:2003.02231 [hep-ph].
  * [11] R. P. Feynman, Phys. Rev. 80 (1950) 440.
  * [12] R. P. Feynman, Phys. Rev. 84 (1951) 108.
  * [13] Z. Bern and D. A. Kosower, Nucl. Phys. B 379 (1992) 451.
  * [14] M. J. Strassler, Nucl. Phys. B 385 (1992) 145, hep-ph/9205205.
  * [15] M. G. Schmidt and C. Schubert, Phys. Lett. B 318 (1993) 438, hep-th/9309055.
  * [16] R. Zh. Shaisultanov, Phys. Lett. B 378 (1996) 354, hep-th/9512142.
  * [17] S.L. Adler and C. Schubert, Phys. Rev. Lett. 77 (1996) 1695, hep-th/9605035.
  * [18] M. Reuter, M. G. Schmidt and C. Schubert, Ann. Phys. (N.Y.) 259 313 (1997), hep-th/9610191.
  * [19] A. Ahmad, N. Ahmadiniaz, O. Corradini, S. P. Kim and C. Schubert, Nucl. Phys. B 919 (2017) 9, arXiv:1612.02944 [hep-ph].
  * [20] N. Ahmadiniaz, F. Bastianelli, O. Corradini, J. P. Edwards and C. Schubert, Nucl. Phys. B 924 (2017) 377, arXiv:1704.05040 [hep-th].
  * [21] C. Schubert, Phys. Rep. 355 (2001) 73, hep-th/0101036.
  * [22] J. P. Edwards and C. Schubert, “Quantum mechanical path integrals in the first quantised approach to quantum field theory”, arXiv:1912.10004 [hep-th].
  * [23] V. I. Ritus, J. of Sov. Laser Res. 6 (1985) 497.
  * [24] F. Ehlotzky, K. Krajewska and J. Z. Kamiński, Rep. Prog. Phys. 72 (2009) 046401.
  * [25] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan and C. H. Keitel, Rev. Mod. Phys. 84 (2012) 1177, arXiv:1111.3886 [hep-ph].
  * [26] A. Ilderton and G. Torgrimsson, Phys. Rev. D 93 (2016) 085006, arXiv:1601.05021 [hep-th].
  * [27] A. M. Polyakov, _Gauge Fields and Strings_ , Harwood (1987).
  * [28] E. S. Fradkin, Nucl. Phys. 76 (1966) 588.
  * [29] N. Ahmadiniaz, V. M. Banda Guzmán, F. Bastianelli, O. Corradini, J. P. Edwards and C. Schubert, JHEP 08 (2020) 049, arXiv:2004.01391 [hep-th].


# On the non primality of certain symmetric ideals

Hyung Kyu Jun  Department of Mathematics
University of Michigan
Ann Arbor, MI 48103
USA<EMAIL_ADDRESS>
###### Abstract.

Let $R=k[x_{1},\cdots,x_{n},\cdots]$ be the infinite variable polynomial ring
equipped with the natural $\mathfrak{S}_{\infty}$ action, where $k$ is a field
of characteristic zero. In recent work [NS21], Nagpal–Snowden gave an indirect
proof that $\mathfrak{S}_{\infty}$-ideal generated by $(x_{1}-x_{2})^{2n}$ is
not $\mathfrak{S}_{\infty}$-prime. In this paper, we give a direct proof, with
explicit elements. We further formulate some conjectures on possible
generalizations of the result.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary linear algebra
  3. 3 Proof of Theorem 1.3
  4. 4 Open Problems

##  1\. Introduction

###  1.1. Background

Let $k$ be a field of characteristic zero, and let $R=k[x_{1},x_{2},\cdots]$
be the polynomial ring in countably many variables $x_{i}$. The ring $R$ is
equipped with a natural action of the infinite symmetric group
$\mathfrak{S}=\bigcup_{n\in\mathbf{N}}\mathfrak{S}_{n}$, where
$\mathfrak{S}_{n}$ is the symmetric group of degree $n$. The ring $R$ is not
noetherian, but the size of $\mathfrak{S}$ bridges the gap: in [Co67] Cohen
showed that $R$ is equivariantly noetherian with respect to $\mathfrak{S}$.
Equivariant noetherianity of $R$ leads us to consider
$\mathfrak{S}$-equivariant commutative algebra on $R$. One is interested in
translating notions from the familiar notions from noetherian rings to a non-
noetherian ring $R$, equipped with an action of $\mathfrak{S}$ (see [NS21] for
basic notions of $G$-equivariant commutative algebra). As prime ideals are one
of the most important notions in commutative algebra and algebraic geometry,
we are interested in $\mathfrak{S}$-prime ideals:

######  Definition 1.1.

An $\mathbf{\mathfrak{S}}$-ideal of $R$ is an ideal closed under the action of
$\mathfrak{S}$. An $\mathfrak{S}$-ideal $\mathfrak{p}$ is
$\mathbf{\mathfrak{S}}$-prime if $f\cdot\sigma(g)\in\mathfrak{p}$ for all
$\sigma\in\mathfrak{S}$ implies $f\in\mathfrak{p}$ or $g\in\mathfrak{p}$.

In fact, $\mathfrak{S}$-prime ideals of $R$ are completely classified in
[NS21], and one of the main steps from the classification of
$\mathfrak{S}$-primes in [NS21] is the following:

######  Theorem 1.2.

For any positive integer $N$, the ideal
$I(N):=\langle(x_{i}-x_{j})^{N}\rangle$ is $\mathfrak{S}$-prime if and only if
$N$ is odd.

Although Theorem 1.2 tells us that $I(2n)$ is not $\mathfrak{S}$-prime, the
proof in [NS21] does not provide $f,g$ such that $f\sigma(g)\in I(2n)$ for all
$\sigma$. The goal of this paper is to provide such an explicit pair.

###  1.2. Main Results

The goal of this paper is to prove the following theorem:

######  Theorem 1.3.

If $f=g=(x_{1}-x_{2})^{2n-1}$, then for any $\sigma\in\mathfrak{S}$, we have
$f\cdot\sigma(g)\in I(2n)$.

Theorem 1.3 implies that $I(2n)$ is not $\mathfrak{S}$-prime for any $n$ as
$(x_{1}-x_{2})^{2n-1}\not\in I(2n)$, because any nonzero element of a
homogeneous ideal $I(2n)$ is of degree $\geq 2n$.

For $n=1$, we can directly show that Theorem 1.3 holds:

######  Example 1.4.

Let $f=g=x_{1}-x_{2},$ and $\sigma\in\mathfrak{S}$ be any permutation. Then,
as any nonzero element of $I(2)$ has degree at least $2$, we see that
$f=g\not\in I(2)$. However, note that $f\cdot\sigma(g)\in I(2)$ from the
following identity:

$(X-Z)^{2}-(Y-Z)^{2}-(X-W)^{2}+(Y-W)^{2}=-2(X-Y)(Z-W).$ (1)

If $\sigma$ maps $x_{1}$ to $x_{i}$ and $x_{2}$ to $x_{j}$, set $X=x_{1}$,
$Y=x_{2}$, $Z=x_{i}$, and $W=x_{j}$, and we see that $f\sigma(g)\in I(2)$ from
(1).

###  1.3. Open problems

In §3, we will see that Theorem 1.3 is related to a contraction of an ideal
$I$ in $k[\delta,x,y]$ to $k[x,y]$. See Theorem 3.6. We then have two
conjectures that generalize Theorem 3.6: Conjecture 3.7 explicitly gives
monomial generators of the contracted ideal, and Conjecture 4.1 generalizes
Theorem 3.6 for more variables. See §4 for details.

###  1.4. Outline

In §3, we give a proof of Theorem 1.3. In §4 we discuss about conjectures that
generalize our main results.

### Acknowledgments

I am grateful to my advisor Andrew Snowden for his support and for numerous
fruitful conversations and suggestions, to Rohit Nagpal for sharing his
thoughts on Schur polynomials and Grobner methods, and David E Speyer for
insights on morphisms involviong triangular matrices.

##  2\. Preliminary linear algebra

In this section, we will discuss two results that will play key roles in the
proof of the Theorem 1.3.

###  2.1. Determinant of a matrix with binomial entries

######  Proposition 2.1.

Let $A$ be the following lower-triangular unipotent $N\times N$ matrix:

$A=\begin{bmatrix}1&0&\cdots&0&0\\\ \\\ {N\choose 1}&1&\cdots&0&0\\\ \\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ \\\ {N\choose N-2}&{N\choose
N-3}&\cdots&1&0\\\ \\\ {N\choose N-1}&{N\choose N-2}&\cdots&{N\choose
1}&1\end{bmatrix}.$ (2)

For $1\leq m\leq N$, let $A_{m}$ be the following lower-left submatrix of $A$

$A_{m}=\begin{bmatrix}{N\choose N-m}&{N\choose N-m-1}&\cdots&0&0\\\ \\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ \\\ {N\choose N-2}&{N\choose
N-3}&\cdots&{N\choose N-m-2}&{N\choose N-m-1}\\\ \\\ {N\choose N-1}&{N\choose
N-2}&\cdots&{N\choose N-m-1}&{N\choose N-m}\end{bmatrix}.$ (3)

Then, $A_{m}$ is an invertible matrix.

###### Proof.

Let $A_{k}$ be the $k\times k$ submatrix of the matrix above. Note that
${n\choose m}=e_{m}(1,\cdots,1)$ where $e_{m}$ is the elementary symmetric
polynomial

$e_{m}(x_{1},\cdots,x_{n}):=\sum_{1\leq i_{1}\leq\cdots\leq i_{m}\leq
n}X_{i_{1}}\cdots X_{i_{m}}.$

Then, we see that the determinant of $A_{k}$ equals the determinant of the
$k\times k$ matrix $B=(b_{i,j})$, evaluated at $(1,\cdots,1)$, where all the
entries $b_{i,j}$ are the elementary symmetric polynomials, corresponding to
$(i,j)^{th}$ entry of $A_{k}$. Using Jacobi–Trudi identity (see [Ma98, pp.
40–41] or [FH13, pp.455]), we see that $\det(B)=s$, where $s$ is a Schur
polynomial. Then the value of $s$ at $(1,\cdots,1)$ is gives the dimension of
the irreducible representation of $GL_{n}$ with highest weight $\lambda$: that
is, if $\lambda=(\lambda_{1}\geq\cdots\geq\lambda_{k}\geq 0)$ and
$s=s_{\lambda}$ is a Schur polynomial associated to the partition $\lambda$,
we have the following closed-form expression

$s_{\lambda}(1,\cdots,1)=\prod_{i<j}\frac{\lambda_{i}-\lambda_{j}+j-i}{j-i}>0.\qed$

######  Remark 2.2.

The above proposition also applies to the matrix whose $(i,j)$th entry is
${n\choose m+i-j}$. The same logic as above applies, so $\det(A)$ is a value
of the corresponding Schur polynomial at $(1,\cdots,1)$. Hence, $\det(A)$ is
positive and in particular, nonzero.

###  2.2. Anti-diagonal transposition

For $n\times n$ matrix $A$, we define $A^{\tau}$ to be an anti-diagonal
transposition,

$A^{\tau}:=JA^{T}J,$

where $J$ is the following matrix:

$J=\begin{bmatrix}0&0&0&\cdots&0&0&1\\\ \\\ 0&0&0&\cdots&0&1&0\\\ \\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ \\\ 0&1&0&\cdots&0&0&0\\\
\\\ 1&0&0&\cdots&0&0&0\end{bmatrix}.$

Note that for any two matrices $A$ and $B$, we have

$(A+B)^{\tau}=A^{\tau}+B^{\tau},$

and

$(AB)^{\tau}=B^{\tau}A^{\tau}.$

We need the following lemma:

######  Lemma 2.3.

If $A=(a_{i,j})_{1\leq i,j\leq n},B=(b_{i,j})_{1\leq i,j\leq n}$ are $n\times
n$ matrices satisfying:

  * •

For any $1\leq m\leq n$, $m\times m$ “upper-right” submatrix

$A_{m}=\begin{bmatrix}a_{1,n-m+1}&a_{1,n-m+2}&\cdots&a_{1,n}\\\
\vdots&\vdots&\ddots&\vdots\\\
a_{m,n-m+1}&a_{m,n+m+2}&\cdots&a_{m,n}\end{bmatrix}$

of $A$ is nonsingular

  * •

All the anti-diagonal entries of $B$ are zero

  * •

$B^{\tau}=-B$.

Then there exists a unique strictly lower-triangular $n\times n$ matrix $X$
satisfying $AX-(AX)^{\tau}=B.$

###### Proof.

Since $X$ is strictly lower triangular matrix, write

$X=\begin{bmatrix}\big{|}&\big{|}&\cdots&\big{|}&\big{|}\\\
X_{1}&X_{2}&\ddots&X_{n-1}&X_{n}\\\
\big{|}&\big{|}&\cdots&\big{|}&\big{|}\end{bmatrix},$

where

$X_{i}=\begin{bmatrix}0\\\ \vdots\\\ 0\\\ x_{i+1,i}\\\ \vdots\\\
x_{n,i}\end{bmatrix}$

is a column vector with $i$ zeros from the top. Specifically, $X_{n}=0$. Also
write

$A=\begin{bmatrix}\text{---}&A_{1}&\text{---}\\\
\text{---}&A_{2}&\text{---}\\\ \vdots&\ddots&\vdots\\\
\text{---}&A_{n}&\text{---}\end{bmatrix},$

where $A_{j}$’s are horizontal $1\times n$ vectors. Then, we see that

$AX=\begin{bmatrix}A_{1}X_{1}&A_{1}X_{2}&\cdots&A_{1}X_{n-1}&0\\\
A_{2}X_{1}&A_{2}X_{2}&\cdots&A_{2}X_{n-1}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
A_{n-1}X_{1}&A_{n-1}X_{2}&\cdots&A_{n-1}X_{n-1}&0\\\
A_{n}X_{1}&A_{n}X_{2}&\cdots&A_{n}X_{n-1}&0\end{bmatrix},$

and

$(AX)^{\tau}=\begin{bmatrix}0&0&\cdots&0&0\\\
A_{n}X_{n-1}&A_{n-1}X_{n-1}&\cdots&A_{2}X_{n-1}&A_{1}X_{n-1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
A_{n}X_{2}&A_{n-1}X_{2}&\cdots&A_{2}X_{2}&A_{1}X_{2}\\\
A_{n}X_{1}&A_{n-1}X_{1}&\cdots&A_{2}X_{1}&A_{1}X_{1}\end{bmatrix},$

Hence, we see that

$AX-(AX)^{\tau}=\begin{bmatrix}A_{1}X_{1}&A_{1}X_{2}&\cdots&A_{1}X_{n-1}&0\\\
A_{2}X_{1}-A_{n}X_{n-1}&A_{2}X_{2}-A_{n-1}X_{n-1}&\cdots&0&-A_{1}X_{n-1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
A_{n-1}X_{1}-A_{n}X_{2}&0&\cdots&-(A_{2}X_{2}-A_{n-1}X_{n-1})&-A_{1}X_{2}\\\
0&-(A_{n-1}X_{1}-A_{n}X_{2})&\cdots&-(A_{2}X_{1}-A_{n}X_{n-1})&-A_{1}X_{1}\end{bmatrix}.$

Thus, if we let

$B=\begin{bmatrix}b_{1,1}&b_{1,2}&\cdots&b_{n-1,n}&0\\\
b_{2,2}&b_{2,2}&\cdots&0&-b_{1,1}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
b_{n-1,1}&0&\cdots&-b_{2,2}&-b_{1,2}\\\
0&-b_{n-1,1}&\cdots&-b_{1,2}&-b_{1,1}\end{bmatrix}$

we only need to check that there exists $X_{1}\cdots X_{n-1}$ so that strictly
upper anti-diagonal entries of $C:=AX-(AX)^{\tau}$ and those of $B$ match.

If we look at $i$th columm($0<i<n$) of $C$ and that of $B$, we require that

$\begin{bmatrix}\text{---}&A_{1}&\text{---}\\\ \vdots&\vdots&\vdots\\\
\text{---}&A_{n-i}&\text{---}\end{bmatrix}\begin{bmatrix}|\\\ X_{i}\\\
|\end{bmatrix}-\begin{bmatrix}\text{---}&0&\text{---}\\\
\vdots&\vdots&\vdots\\\ \text{---}&0&\text{---}\\\
\text{---}&A_{n-i+1}&\text{---}\end{bmatrix}\begin{bmatrix}|\\\ X_{i+1}\\\
|\end{bmatrix}+\cdots+\begin{bmatrix}\text{---}&A_{n-i+1}&\text{---}\\\
\text{---}&0&\text{---}\\\ \vdots&\vdots&\vdots\\\
\text{---}&0&\text{---}\end{bmatrix}\begin{bmatrix}|\\\ X_{n-1}\\\
|\end{bmatrix}=\begin{bmatrix}b_{1,i}\\\ b_{2,i}\\\ \vdots\\\
b_{n-i,i}\end{bmatrix}$

Here, recall that $X_{i}$ is a colum vector with $i$ consecutive zeros from
the top. Hence,

$\begin{bmatrix}\text{---}&A_{1}&\text{---}\\\ \vdots&\vdots&\vdots\\\
\text{---}&A_{n-i}&\text{---}\end{bmatrix}\begin{bmatrix}|\\\ X_{i}\\\
|\end{bmatrix}=\begin{bmatrix}\mathbb{0}&\vline&A_{i},\end{bmatrix}\begin{bmatrix}|\\\
X_{i}\\\ |\end{bmatrix}$

where $A_{i}$ is $i\times i$ upper right submatrix of $A$. Hence, if we write
$X_{i}^{\prime}$ to be $(n-i)\times 1$ column vector obtained by removing
consecutive zeros from $X_{i}$, we see that (in a block matrix form),

$\begin{bmatrix}A_{n-1}&*&*&\cdots&*&*\\\ 0&A_{n-2}&*&\cdots&*&*\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&0&A_{1}\end{bmatrix}\begin{bmatrix}X_{1}^{\prime}\\\
X_{2}^{\prime}\\\ \vdots\\\ X_{n-1}^{\prime}\end{bmatrix}=\vec{b},$

where $\vec{b}$ is $n(n-1)/2\times 1$ column vector consisting of strictly
upper anti-diagonal entries from $B$. Note that the coefficient matrix of the
above equation is nonsingular, since its determinant equals to
$\det(A_{1})\cdot\cdots\cdot\det(A_{n-1}),$ which is nonzero by assumption.
This completes the proof. ∎

##  3\. Proof of Theorem 1.3

###  3.1. Suitable change of variables

Let $x_{i,j}:=x_{i}-x_{j}$. We show the following is sufficient to prove
Theorem 1.3:

######  Proposition 3.1.

We have

$(x_{1,2})^{2n-1}(x_{3,4})^{2n-1}\in\langle(x_{1,3})^{2n},(x_{1,4})^{2n},(x_{2,3})^{2n},(x_{2,4})^{2n}\rangle.$

###### Proof of Theorem 1.3, assuming Proposition 3.1.

If $\sigma$ sends $x_{1}\to x_{i}$ and $x_{2}\to x_{j}$, set $x_{3}=x_{i}$,
and $x_{4}=x_{j}$. Then we see that

$f\cdot\sigma(g)=(x_{1,2})^{2n-1}(x_{i,j})^{2n-1}\in\langle(x_{1,i})^{2n},(x_{1,j})^{2n},(x_{2,i})^{2n},(x_{2,j})^{2n}\rangle\subset
I(2n).$

Hence $f\cdot\sigma(g)\in I(2n)$ for all $\sigma$. ∎

To prove the above proposition, we will make a change of variables. Set
$S=k[\delta,x,y]$, and define

$J(2n):=\langle\delta^{2n},(\delta+x)^{2n},(\delta+y)^{2n},(\delta+x+y)^{2n}\rangle.$

If we write $x=x_{1,2}$, $y=-x_{3,4}$, and $\delta=x_{2,4}$, we see that

$(x_{1,2})^{2n-1}(x_{3,4})^{2n-1}=-(xy)^{2n-1},$

and

$\langle(x_{1,3})^{2n},(x_{1,4})^{2n},(x_{2,3})^{2n},(x_{2,4})^{2n}\rangle=J(2n).$

Hence, Proposition 3.1 follows from the following lemma:

######  Lemma 3.2.

We have $(xy)^{2n-1}\in J(2n)$.

###  3.2. The Proof

Finally, we will complete the proof of Theorem 1.3 by showing Lemma 3.2.

###### Proof of Lemma 3.2.

We need to find homogeneous polynomials
$f_{\delta},f_{\delta+x},f_{\delta+y},f_{\delta+x+y}\in k[\delta,x,y]$ of
degree $2n-2$ such that

$(xy)^{2n-1}=f_{\delta}\delta^{2n}+f_{\delta+x}(\delta+x)^{2n}+f_{\delta+y}(\delta+y)^{2n}+f_{\delta+x+y}(\delta+x+y)^{2n}$
(4)

Divide both sides by $\delta^{4n-2}$. We then get

$\left(\frac{x}{\delta}\right)^{2n-1}\cdot\left(\frac{y}{\delta}\right)^{2n-1}=f_{\delta}^{\prime}+f_{\delta+x}^{\prime}\cdot\left(1+\frac{x}{\delta}\right)^{2n}+f^{\prime}_{\delta+y}\cdot\left(1+\frac{y}{\delta}\right)^{2n}+f_{\delta+x+y}^{\prime}\cdot\left(1+\frac{x}{\delta}+\frac{y}{\delta}\right)^{2n},$

where $f_{*}^{\prime}$ are polynomials of degree at most $2n-2$ obtained by
dehomogenization of $f_{*}$’s:

$f_{*}^{\prime}\left(\frac{x}{\delta},\frac{y}{\delta}\right)=f_{*}\left(1,\frac{x}{\delta},\frac{y}{\delta}\right)$

We will find the equations between the coefficients, and although such
comparison is possible without dehomogenization, it is more convenient to
eliminate one of the variables. Then set $s=1+x/\delta$, and $t=y/\delta$.
Define $a,b,c,d$ to be polynomials in $s$ and $t$ obtained from
$f_{\delta}^{\prime},\cdots,f_{\delta+x+y}^{\prime}$ after the change of
variables. Then, we are reduced to show that there exist polynomials $a,b,c,d$
in $s,t$ with degrees $\leq 2n-2$ such that

$(s-1)^{2n-1}t^{2n-1}=a+b\cdot s^{2n}+c\cdot(1+t)^{2n}+d\cdot(s+t)^{2n}.$ (5)

Note that we have two gradings, obtained by dehomogenization: one from
$s$–degree and another from $t$-degree. Such will allow us to obtain the
relationships between the coefficients of the terms, and we will show that
there exist a (unique) pair of coefficients that satisfy the relationships
between the coefficients of $a,b,c,d$ that we will soon derive.

Put

$b=\sum_{i=0}^{2n-2}b_{i}s^{i},$

where $b_{i}=b_{i}(t)$ is a polynomial in $t$ with degree $\leq i$ for $0\leq
i\leq 2n-2$. Similarly write $a=\sum a_{i}s^{i}$, $c=\sum c_{i}s^{i}$, and
$d=\sum d_{i}s^{i}$. Expand the term $d(s+t)^{2n}$ in (5), and compare the
coefficients of $s^{k}$: i.e., polynomial in $t$\- from $k=0$ to $k=4n-2$.

We now limit our attention to terms with $s$–degree $\geq 2n$. Observe that
such terms can only appear from $b\cdot s^{2n}$ and $d\cdot(s+t)^{2n}$. As the
terms from the expansion $c\cdot(1+t)^{2n}$ cannot have such terms as the
degrees of $b$ (and thus $s$–degrees of those) is at most $2n-2$. Thus, we get

$\displaystyle d_{2n-2}t^{2n-2}{2n\choose 2n-2}+\cdots+d_{1}t{2n\choose
1}+d_{0}$ $\displaystyle=-b_{0}\text{ }(\text{for }s^{2n})$
$\displaystyle\text{ }\vdots$ $\displaystyle d_{2n-2}t{2n\choose 1}+d_{2n-3}$
$\displaystyle=-b_{2n-3}\text{ }(\text{for }s^{4n-3})$ $\displaystyle
d_{2n-2}$ $\displaystyle=-b_{2n-2}\text{ }(\text{for }s^{4n-2}).$

Multiply the first equation by $1/(s^{2n})$, thesecond by $t/s^{2n+1}$,
$\cdots$, and the last equation by $t^{2n}/s^{4n-2}$. We then get the
following equation:

$\begin{bmatrix}1&{2n\choose 1}&\cdots&{2n\choose 2n-2}\\\ \\\
0&1&\cdots&{2n\choose 2n-3}\\\ \\\ \vdots&\vdots&\vdots&\vdots\\\ \\\
0&0&\cdots&1\end{bmatrix}\cdot\begin{bmatrix}d_{0}\\\ \\\ d_{1}t\\\ \\\
\vdots\\\ \\\ d_{2n-2}t^{2n-2}\end{bmatrix}=-\begin{bmatrix}b_{0}\\\ \\\
b_{1}t\\\ \\\ \vdots\\\ \\\ b_{2n-2}t^{2n-2}\end{bmatrix}.$ (6)

Thus, we see that the $d$’s uniquely determine the $b$’s.

Now, compare the terms with $s$-degree from zero to to $2n-1$ from the
equation (5). Now the terms we have come from the remaining terms of
$d\cdot(s+t)^{2n}$, from $a$ and $c\cdot(1+t)^{2n}$, and lastly from the
expansion of $(s-1)^{2n-1}t^{2n-1},$ where the coefficients are entirely
determined by binomial coefficients. We write the terms in that order:

$\displaystyle d_{0}t^{2n}$
$\displaystyle=-a_{0}-c_{0}(1+t)^{2n}+(-1)^{2n-1}{2n-1\choose 2n-1}t^{2n-1}$
$\displaystyle\sum_{i=2n-1}^{2n}d_{i-(2n-1)}{2n\choose i}t^{i}$
$\displaystyle=-a_{1}-c_{1}(1+t)^{2n}+(-1)^{2n-2}{2n-1\choose 2n-2}t^{2n-1}$
$\displaystyle\vdots$ (7) $\displaystyle\sum_{i=2}^{2n}d_{i-2}{2n\choose
i}t^{i}$ $\displaystyle=-a_{2n-2}-c_{2n-2}(1+t)^{2n}+(-1)^{1}{2n-1\choose
1}t^{2n-1}$ $\displaystyle\sum_{i=1}^{2n-1}d_{i-1}{2n\choose i}t^{i}$
$\displaystyle=-0-0+(-1)^{0}{2n-1\choose 0}t^{2n-1}$

Write

$c_{k}=c_{k,0}+\cdots+c_{k,2n-2-k}t^{2n-2-k},\text{ and
}d_{k}=d_{k,0}+d_{k,1}t+\cdots+d_{k,2n-2-k}t^{2n-2-k}.$

Here, note that $c_{k}$ and $d_{k}$ are polynomials in $t$ of degree $2n-2-k$,
and $c_{i,j}$ and $d_{i,j}$ are scalar coefficients, of polynomials $c_{i}$
and $d_{i}$ respectively. Recall that $\deg a_{i}\leq i$, so $a_{i}$’s can be
thought as the remainder of $(RHS)-(LHS)$ with respect to $t^{i}$. Hence, if
we consider the quotient of $(LHS)$ and $(RHS)$ of the equation (3.2) with
respect to the divisor $t^{i}$, we get the following equations system of
$2n-2$ linear equations involving $c_{ij}$’s and $d_{ij}$’s:

$\Lambda C+D\Lambda=B.$ (8)

Here, $\Lambda$ is an upper triangular unipotent matrix

$\Lambda=\begin{bmatrix}1&{2n\choose 1}&{2n\choose 2}&\cdots&{2n\choose
2n-3}&{2n\choose 2n-2}\\\ \\\ 0&1&{2n\choose 1}&\cdots&{2n\choose
2n-4}&{2n\choose 2n-3}\\\ \\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ \\\
0&0&0&\cdots&1&{2n\choose 1}\\\ \\\ 0&0&0&\cdots&0&1\end{bmatrix},$ (9)

and $B$ is a diagonal matrix

$B=\textbf{Diag}\left[(-1)^{2n-1}{2n-1\choose 2n-1},(-1)^{2n-2}{2n-1\choose
2n-2},\cdots,(-1)^{0}{2n-1\choose 0}\right],$ (10)

and $C$, $D$ are the following strictly lower trianglular matrices

$C,D=\begin{bmatrix}0&0&0&\cdots&0\\\ \\\ c_{0,0}&0&0&\cdots&0\\\ \\\
c_{0,1}&c_{1,0}&0&\cdots&0\\\ \\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\ \\\
c_{0,2n-2}&c_{1,2n-3}&c_{2,2n-4}&\cdots&c_{2n-2,0}\end{bmatrix},\begin{bmatrix}0&0&0&\cdots&0\\\
\\\ d_{0,0}&0&0&\cdots&0\\\ \\\ d_{0,1}&d_{1,0}&0&\cdots&0\\\ \\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \\\
d_{0,2n-2}&d_{1,2n-3}&d_{2,2n-4}&\cdots&d_{2n-2,0}\end{bmatrix}.$ (11)

Using Proposition 2.1, we see that $\Lambda,B$ satisfies the condition of
Lemma 2.3. Hence by Lemma 2.3, there is a pair $(C,D)$ with $D=-C^{\tau}$ that
satisfies (8). This completes the proof. ∎

######  Remark 3.3.

Conceptually, the reason why we expect an equation symmetric in $C$ (the
coefficient matrix of $c$) and $D$(those of $d$) is that once we apply
different change of variables and dehomogenizations to (4), we can actually
interchange $c$ and $d$. Consider the (well-defined linear) map $\phi$ induced
by

$\displaystyle\delta$ $\displaystyle\mapsto\delta+x,$ (12)
$\displaystyle\delta+x$ $\displaystyle\mapsto\delta,$ (13)
$\displaystyle\delta+y$ $\displaystyle\mapsto\delta+x+y,$ (14)
$\displaystyle\delta+x+y$ $\displaystyle\mapsto\delta+y,$ (15)

and dehomogenize after applying $\phi$. Then, $c$ and $d$ are mapped to $d$
and $c$. As a consequence we expect to observe some symmetry, i.e., one that
we see from (8).

######  Example 3.4.

By unraveling, we can compute explicit polynomials $a,b,c,d$ satisfying
$(xy)^{2n-1}=a\delta^{2n}+\cdots+d(\delta+x+y)^{2n}$ for any specific values
of $n$. For instance, when $2n=4$, we have

$\begin{bmatrix}a(\delta,x,y)\\\ \\\ b(\delta,x,y)\\\ \\\ c(\delta,x,y)\\\ \\\
d(\delta,x,y)\end{bmatrix}=\begin{bmatrix}\frac{1}{5}\delta^{2}+\frac{3}{5}\delta
x+\frac{3}{5}\delta y+\frac{3}{2}xy\\\ \\\
-\frac{1}{5}\delta^{2}+\frac{1}{5}\delta x+\frac{2}{5}x^{2}-\frac{3}{5}\delta
y+\frac{9}{10}xy\\\ \\\ -\frac{1}{5}\delta^{2}-\frac{3}{5}\delta
x+\frac{1}{5}\delta y+\frac{9}{10}xy+\frac{2}{5}y^{2}\\\ \\\
\frac{1}{5}\delta^{2}-\frac{1}{5}\delta x-\frac{2}{5}x^{2}-\frac{1}{5}\delta
y+\frac{7}{10}xy-\frac{2}{5}y^{2}\end{bmatrix},$

and we can check that indeed,

$a\delta^{4}+b(\delta+x)^{4}+c(\delta+y)^{4}+d(\delta+x+y)^{4}=(xy)^{3}.$

######  Remark 3.5.

We can replace $k$, a field of characteristic zero, with any field of
characteristic $p$ such that the statement of Theorem 2.1 holds.

In fact, we can show the following general result:

######  Theorem 3.6.

Let $N$ be a positive integer. Then the contracted ideal
$\mathfrak{J}(N):=J(N)\cap k[x,y]$ contains $\langle
x^{2N-1},y^{2N-1},x^{2i+1}y^{2j+1}|i+j=N-2\rangle$.

###### Proof.

Without loss of generality assume $i\leq j$. We can show that
$x^{2i+1}y^{2j+1}\in J(N)$ with the same dehomogenization and comparison of
coefficients, and we see that

$x^{2N-1}=((\delta+x)-\delta)^{2N-1}\in\langle\delta^{N},(\delta+x)^{N}\rangle$

from binomial expansion. ∎

Note that Theorem 1.3 is a special case of Theorem 3.6. Furthermore, we have a
following conjecture that the contracted ideal $\mathfrak{J}(N)$ is actually a
monomial ideal:

######  Conjecture 3.7.

In fact, $\mathfrak{J}(N)$ equals to $\langle
x^{2N-1},y^{2N-1},x^{2i+1}y^{2j+1}|i+j=N-2\rangle$.

##  4\. Open Problems

In this section, we give more conjectures that generalize Theorem 1.3 and
Theorem 3.6.

###  4.1. More variables

Fix $r,n\in\mathbf{N}$, and for a subset $S$ of $[r]:=\\{1,2,\cdots,r\\}$,
define $t_{S}=\sum_{i\in S}t_{i}$, and $t_{S,0}=t_{0}+t_{S}$. Here, $t_{0}$ is
a special variable, that takes a role of $\delta$ in (4). We then define
$I_{r}^{(d)}\subset R:=K[t_{0},\cdots,t_{r}]$ to be

$I_{r}^{(d)}:=\langle(t_{S,0})^{d}:S\subset[r]\rangle.$

######  Conjecture 4.1.

For any $r\geq 2$ and $n\geq 1$, we have

$\left(t_{1}\cdots t_{r}\right)^{2n-1}\in I_{r}^{(nr)}.$

For $r=1$, the above conjecture reduces to $t_{1}^{2n-1}\in\langle
t_{0}^{n},(t_{0}+t_{1})^{n}\rangle,$ which follows easily from the binomial
expansion of $t^{2n-1}=((t+\delta)-\delta)^{2n-1}.$ Our theorem gives a
positive answer for $r=2$. We have verified the conjecture for some small
values of $r\geq 3$ by computer.

###  4.2. Betti tables

We also observe some patterns from Betti tables of the family of ideals
$I_{r}^{(d)}$ with fixed $r$. Let $\mathbf{F}$ be the minimal free resolution
of $I_{r}^{(d)}$, and write

$F_{i}=\bigoplus_{j\in\mathbf{Z}}R(-j)^{\beta_{i,j}}.$

When $r=1$, we expect $\beta_{i,j}=0$ for all $i,j$ except for
$(i,j)=(0,0),(d-1,1),$ and $(2d-2,2)$. The nonzero values of $\beta_{i,j}$’s
are given by $\beta_{0,0}=\beta_{2d-2,2}=1,\beta_{d-1,1}=2$. See the following
Betti table, where all the omitted rows are zero rows, and entries with
$\cdot$ is a zero entry:

0 1 2 0 1 $\cdot$ $\cdot$ ⋮⋮⋮⋮$d-1$ $\cdot$ 2 $\cdot$ ⋮⋮⋮⋮$2d-2$ $\cdot$
$\cdot$ 1 ij

For For $r=2$, we have the following conjectured Betti table obtained for some
small values we tested by computer:

0 1 2 3 0 1 $\cdot$ $\cdot$ $\cdot$ ⋮⋮⋮⋮⋮$d-1$$\cdot$ 4 $\cdot$ $\cdot$
⋮⋮⋮⋮⋮$2d-3$$\cdot$ $\cdot$ $d$ $\cdot$ $2d-2$$\cdot$ $\cdot$ 3$d$ ij

Lastly, we give the Betti table for $r=3$:

0 1 2 3 40 1 $\cdot$ $\cdot$ $\cdot$ $\cdot$⋮⋮⋮⋮⋮⋮$d-1$$\cdot$ 8 $\cdot$
$\cdot$ $\cdot$⋮⋮⋮⋮⋮⋮$2d-4$$\cdot$ $\cdot$ $(d-1)d/2$ $\cdot$ $\cdot$
$2d-3$$\cdot$ $\cdot$ $4d$ $(d-1)(d+1)$ $\cdot$ $2d-2$$\cdot$ $\cdot$ 6$4d$
$d(d-1)/2$ ij

There are a number of patterns we can observe: why do we expect each entries
of Betti table to stabilize? For instance, why do we expect to get “6” at the
lower left vertex of the triangle (i.e., $\beta_{2d-2,3}=6$ for $r=3$)? More
fundamentally, why do we see $r\times r$ right triangle at the lower right
corner? Should we expect to observe similar patterns for larger values of $r$?

## References

  * [Co67] D. E. Cohen. “On the laws of a metabelian variety.” J. Algebra 5 (1967), 267-273.
  * [Co87] D. E. Cohen. “Closure relations, Buchberger’s algorithm, and polynomials in infinitely many variables.” In Computation theory and logic, volume 270 of Lect. Notes Comput. Sci., pp. 78-87, 1987.
  * [Ful97] W. Fulton. “ Young tableaux: with applications to representation theory and geometry”. Cambridge University Press, 1997.
  * [FH13] W. Fulton and J. Harris. “Representation theory: a first course”. Springer, 2013.
  * [Ma98] I. G. Macdonald. “Symmetric functions and Hall polynomials”. Oxford university press, 1998.
  * [NS21] R. Nagpal and A. Snowden. “Symmetric ideals of the infinite polynomial ring” Preprint, 2021 arXiv:2107.13027


Minerviniorg104,org23&A. Mischkeorg57&A.N. Mishraorg49&D. Miśkowiecorg97&J.
Mitraorg131&C.M. Mituorg62&N. Mohammadiorg57&B. Mohantyorg79,org131&L.
Molnarorg55&L. Montaño Zetinaorg11&E. Montesorg10&M. Morandoorg30&S.
Morettoorg30&A. Morrealeorg113&A. Morschorg36&V. Mucciforaorg72&E.
Mudnicorg115&D. Mühlheimorg54&S. Muhuriorg131&M. Mukherjeeorg131&H.
Müllerorg36&J.D. Mulliganorg136&M.G. Munhozorg119&S. Murrayorg65&L.
Musaorg36&J. Musinskyorg59&B.K. Nandiorg48&R. Naniaorg105&E. Nappiorg104&M.U.
Naruorg16&C. Nattrassorg124&K. Nayakorg79&T.K. Nayakorg131&S.
Nazarenkoorg99&A. Nedosekinorg58&L. Nellenorg63&F. Ngorg121&M.
Nicassioorg97&M. Niculescuorg36,org62&J. Niedzielaorg36&B.S. Nielsenorg80&S.
Nikolaevorg100&S. Nikulinorg100&V. Nikulinorg85&F. Noferiniorg105,org12&P.
Nomokonovorg66&G. Noorenorg57&J. Normanorg123&A. Nyaninorg100&J.
Nystrandorg18&H. Oeschlerorg93&S. Ohorg136&S.K. Ohorg67&A. Ohlsonorg36&A.
Okatanorg69&T. Okuboorg47&L. Olahorg135&J. Oleniaczorg133&A.C. Oliveira Da
Silvaorg119&M.H. Oliverorg136&J. Onderwaaterorg97&C. Oppedisanoorg111&A. Ortiz
Velasquezorg63&A. Oskarssonorg34&J. Otwinowskiorg97,org116&K. Oyamaorg93&M.
Ozdemirorg53&Y. Pachmayerorg93&P. Paganoorg31&G. Paićorg63&C.
Pajaresorg17&S.K. Palorg131&J. Panorg134&A.K. Pandeyorg48&D. Pantorg48&V.
Papikyanorg1&G.S. Pappalardoorg107&P. Pareekorg49&W.J. Parkorg97&S.
Parmarorg87&A. Passfeldorg54&V. Paticchioorg104&B. Paulorg101&T.
Pawlakorg133&T. Peitzmannorg57&H. Pereira Da Costaorg15&E. Pereira De Oliveira
Filhoorg119&D. Peresunkoorg76,org100&C.E. Pérez Laraorg81&V. Peskovorg53&Y.
Pestovorg5&V. Petráčekorg40&V. Petrovorg112&M. Petroviciorg78&C. Pettaorg29&S.
Pianoorg110&M. Piknaorg39&P. Pillotorg113&O. Pinazzaorg105,org36&L.
Pinskyorg121&D.B. Piyarathnaorg121&M. Płoskońorg74&M. Planinicorg128&J.
Plutaorg133&S. Pochybovaorg135&P.L.M. Podesta-Lermaorg118&M.G.
Poghosyanorg86&B. Polichtchoukorg112&N. Poljakorg128&W. Poonsawatorg114&A.
Poporg78&S. Porteboeuf-Houssaisorg70&J. Porterorg74&J. Pospisilorg83&S.K.
Prasadorg4&R. Preghenellaorg105,org36&F. Prinoorg111&C.A. Pruneauorg134&I.
Pshenichnovorg56&M. Puccioorg111&G. Pudduorg25&P. Pujahariorg134&V.
Puninorg99&J. Putschkeorg134&H. Qvigstadorg22&A. Rachevskiorg110&S.
Rahaorg4&S. Rajputorg90&J. Rakorg122&A. Rakotozafindrabeorg15&L.
Ramelloorg32&R. Raniwalaorg91&S. Raniwalaorg91&S.S. Räsänenorg46&B.T.
Rascanuorg53&D. Ratheeorg87&V. Razaziorg25&K.F. Readorg124&J.S. Realorg71&K.
Redlichorg77&R.J. Reedorg134&A. Rehmanorg18&P. Reicheltorg53&M.
Reicherorg57&F. Reidtorg93,org36&X. Renorg7&R. Renfordtorg53&A.R.
Reolonorg72&A. Reshetinorg56&F. Rettigorg43&J.-P. Revolorg12&K.
Reygersorg93&V. Riabovorg85&R.A. Ricciorg73&T. Richertorg34&M. Richterorg22&P.
Riedlerorg36&W. Rieglerorg36&F. Riggiorg29&C. Risteaorg62&A. Rivettiorg111&E.
Roccoorg57&M. Rodríguez Cahuantziorg11,org2&A. Rodriguez Mansoorg81&K.
Røedorg22&E. Rogochayaorg66&D. Rohrorg43&D. Röhrichorg18&R. Romitaorg123&F.
Ronchettiorg72&L. Ronfletteorg113&P. Rosnetorg70&A. Rossiorg36&F.
Roukoutakisorg88&A. Royorg49&C. Royorg55&P. Royorg101&A.J. Rubio
Monteroorg10&R. Ruiorg26&R. Russoorg27&E. Ryabinkinorg100&Y. Ryabovorg85&A.
Rybickiorg116&S. Sadovskyorg112&K. Šafaříkorg36&B. Sahlmullerorg53&P.
Sahooorg49&R. Sahooorg49&S. Sahooorg61&P.K. Sahuorg61&J. Sainiorg131&S.
Sakaiorg72&M.A. Salehorg134&C.A. Salgadoorg17&J. Salzwedelorg20&S.
Sambyalorg90&V. Samsonovorg85&X. Sanchez Castroorg55&L. Šándororg59&A.
Sandovalorg64&M. Sanoorg127&G. Santagatiorg29&D. Sarkarorg131&E.
Scapparoneorg105&F. Scarlassaraorg30&R.P. Scharenbergorg95&C. Schiauaorg78&R.
Schickerorg93&C. Schmidtorg97&H.R. Schmidtorg35&S. Schuchmannorg53&J.
Schukraftorg36&M. Schulcorg40&T. Schusterorg136&Y. Schutzorg113,org36&K.
Schwarzorg97&K. Schwedaorg97&G. Scioliorg28&E. Scomparinorg111&R.
Scottorg124&K.S. Seederorg119&J.E. Segerorg86&Y. Sekiguchiorg126&I.
Selyuzhenkovorg97&K. Senosiorg65&J. Seoorg67,org96&E. Serradillaorg10,org64&A.
Sevcencoorg62&A. Shabanovorg56&A. Shabetaiorg113&O. Shaduraorg3&R.
Shahoyanorg36&A. Shangaraevorg112&A. Sharmaorg90&M. Sharmaorg90&N.
Sharmaorg61,org124&K. Shigakiorg47&K. Shtejerorg27,org9&Y. Sibiriakorg100&S.
Siddhantaorg106&K.M. Sielewiczorg36&T. Siemiarczukorg77&D.
Silvermyrorg84,org34&C. Silvestreorg71&G. Simatovicorg128&G. Simonettiorg36&R.
Singarajuorg131&R. Singhorg90,org79&S. Singhaorg79,org131&V.
Singhalorg131&B.C. Sinhaorg131&T. Sinhaorg101&B. Sitarorg39&M. Sittaorg32&T.B.
Skaaliorg22&M. Slupeckiorg122&N. Smirnovorg136&R.J.M. Snellingsorg57&T.W.
Snellmanorg122&C. Søgaardorg34&R. Soltzorg75&J. Songorg96&M. Songorg137&Z.
Songorg7&F. Soramelorg30&S. Sorensenorg124&M. Spacekorg40&E. Spiritiorg72&I.
Sputowskaorg116&M. Spyropoulou-Stassinakiorg88&B.K. Srivastavaorg95&J.
Stachelorg93&I. Stanorg62&G. Stefanekorg77&M. Steinpreisorg20&E.
Stenlundorg34&G. Steynorg65&J.H. Stillerorg93&D. Stoccoorg113&P.
Strmenorg39&A.A.P. Suaideorg119&T. Sugitateorg47&C. Suireorg51&M.
Suleymanovorg16&R. Sultanovorg58&M. Šumberaorg83&T.J.M. Symonsorg74&A.
Szaboorg39&A. Szanto de Toledoorg1190&I. Szarkaorg39&A.
Szczepankiewiczorg36&M. Szymanskiorg133&J. Takahashiorg120&N.
Tanakaorg127&M.A. Tangaroorg33&J.D. Tapia Takakiidp5866224,org51&A. Tarantola
Peloniorg53&M. Tariqorg19&M.G. Tarzilaorg78&A. Tauroorg36&G. Tejeda
Muñozorg2&A. Telescaorg36&K. Terasakiorg126&C. Terrevoliorg30,org25&B.
Teyssierorg129&J. Thäderorg97,org74&D. Thomasorg57,org117&R.
Tieulentorg129&A.R. Timminsorg121&A. Toiaorg53&S. Trogoloorg111&V.
Trubnikovorg3&W.H. Trzaskaorg122&T. Tsujiorg126&A. Tumkinorg99&R.
Turrisiorg108&T.S. Tveterorg22&K. Ullalandorg18&A. Urasorg129&G.L.
Usaiorg25&A. Utrobicicorg128&M. Vajzerorg83&M. Valaorg59&L. Valencia
Palomoorg70&S. Valleroorg27&J. Van Der Maarelorg57&J.W. Van Hoorneorg36&M. van
Leeuwenorg57&T. Vanatorg83&P. Vande Vyvreorg36&D. Vargaorg135&A. Vargasorg2&M.
Vargyasorg122&R. Varmaorg48&M. Vasileiouorg88&A. Vasilievorg100&A.
Vauthierorg71&V. Vecherninorg130&A.M. Veenorg57&M. Veldhoenorg57&A.
Velureorg18&M. Venaruzzoorg73&E. Vercellinorg27&S. Vergara Limónorg2&R.
Vernetorg8&M. Verweijorg134&L. Vickovicorg115&G. Viestiorg300&J.
Viinikainenorg122&Z. Vilakaziorg125&O. Villalobos Baillieorg102&A.
Vinogradovorg100&L. Vinogradovorg130&Y. Vinogradovorg990&T. Virgiliorg31&V.
Vislaviciusorg34&Y.P. Viyogiorg131&A. Vodopyanovorg66&M.A. Völklorg93&K.
Voloshinorg58&S.A. Voloshinorg134&G. Volpeorg135,org36&B. von Hallerorg36&I.
Vorobyevorg92,org37&D. Vranicorg97,org36&J. Vrlákováorg41&B. Vulpescuorg70&A.
Vyushinorg99&B. Wagnerorg18&J. Wagnerorg97&H. Wangorg57&M. Wangorg7,org113&Y.
Wangorg93&D. Watanabeorg127&M. Weberorg36,org121&S.G. Weberorg97&J.P.
Wesselsorg54&U. Westerhofforg54&J. Wiechulaorg35&J. Wikneorg22&M.
Wildeorg54&G. Wilkorg77&J. Wilkinsonorg93&M.C.S. Williamsorg105&B.
Windelbandorg93&M. Winnorg93&C.G. Yaldoorg134&Y. Yamaguchiorg126&H.
Yangorg57&P. Yangorg7&S. Yanoorg47&S. Yasnopolskiyorg100&Z. Yinorg7&H.
Yokoyamaorg127&I.-K. Yooorg96&V. Yurchenkoorg3&I. Yushmanovorg100&A.
Zaborowskaorg133&V. Zaccoloorg80&A. Zamanorg16&C. Zampolliorg105&H.J.C.
Zanoliorg119&S. Zaporozhetsorg66&A. Zarochentsevorg130&P. Závadaorg60&N.
Zaviyalovorg99&H. Zbroszczykorg133&I.S. Zguraorg62&M. Zhalovorg85&H.
Zhangorg18,org7&X. Zhangorg74&Y. Zhangorg7&C. Zhaoorg22&N. Zhigarevaorg58&D.
Zhouorg7&Y. Zhouorg57&Z. Zhouorg18&H. Zhuorg7,org18&J. Zhuorg7,org113&X.
Zhuorg7&A. Zichichiorg12,org28&A. Zimmermannorg93&M.B.
Zimmermannorg54,org36&G. Zinovjevorg3&M. Zyzakorg43

## Affiliation notes

0Deceased

idp3748880Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn
Institute of Nuclear, Physics, Moscow, Russia

idp5866224Also at: University of Kansas, Lawrence, Kansas, United States

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

org4Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org5Budker Institute for Nuclear Physics, Novosibirsk, Russia

org6California Polytechnic State University, San Luis Obispo, California,
United States

org7Central China Normal University, Wuhan, China

org8Centre de Calcul de l’IN2P3, Villeurbanne, France

org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain

org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi”, Rome, Italy

org13Chicago State University, Chicago, Illinois, USA

org14China Institute of Atomic Energy, Beijing, China

org15Commissariat à l’Energie Atomique, IRFU, Saclay, France

org16COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan

org17Departamento de Física de Partículas and IGFAE, Universidad de Santiago
de Compostela, Santiago de Compostela, Spain

org18Department of Physics and Technology, University of Bergen, Bergen,
Norway

org19Department of Physics, Aligarh Muslim University, Aligarh, India

org20Department of Physics, Ohio State University, Columbus, Ohio, United
States

org21Department of Physics, Sejong University, Seoul, South Korea

org22Department of Physics, University of Oslo, Oslo, Norway

org23Dipartimento di Elettrotecnica ed Elettronica del Politecnico, Bari,
Italy

org24Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN
Rome, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org26Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org27Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org30Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org31Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy

org33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org34Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden

org35Eberhard Karls Universität Tübingen, Tübingen, Germany

org36European Organization for Nuclear Research (CERN), Geneva, Switzerland

org37Excellence Cluster Universe, Technische Universität München, Munich,
Germany

org38Faculty of Engineering, Bergen University College, Bergen, Norway

org39Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia

org40Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org41Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org42Faculty of Technology, Buskerud and Vestfold University College,
Vestfold, Norway

org43Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org44Gangneung-Wonju National University, Gangneung, South Korea

org45Gauhati University, Department of Physics, Guwahati, India

org46Helsinki Institute of Physics (HIP), Helsinki, Finland

org47Hiroshima University, Hiroshima, Japan

org48Indian Institute of Technology Bombay (IIT), Mumbai, India

org49Indian Institute of Technology Indore, Indore (IITI), India

org50Inha University, Incheon, South Korea

org51Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud,
CNRS-IN2P3, Orsay, France

org52Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org53Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org54Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
Münster, Germany

org55Institut Pluridisciplinaire Hubert Curien (IPHC), Université de
Strasbourg, CNRS-IN2P3, Strasbourg, France

org56Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org57Institute for Subatomic Physics of Utrecht University, Utrecht,
Netherlands

org58Institute for Theoretical and Experimental Physics, Moscow, Russia

org59Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org60Institute of Physics, Academy of Sciences of the Czech Republic, Prague,
Czech Republic

org61Institute of Physics, Bhubaneswar, India

org62Institute of Space Science (ISS), Bucharest, Romania

org63Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org64Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org65iThemba LABS, National Research Foundation, Somerset West, South Africa

org66Joint Institute for Nuclear Research (JINR), Dubna, Russia

org67Konkuk University, Seoul, South Korea

org68Korea Institute of Science and Technology Information, Daejeon, South
Korea

org69KTO Karatay University, Konya, Turkey

org70Laboratoire de Physique Corpusculaire (LPC), Clermont Université,
Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France

org71Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org72Laboratori Nazionali di Frascati, INFN, Frascati, Italy

org73Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy

org74Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org75Lawrence Livermore National Laboratory, Livermore, California, United
States

org76Moscow Engineering Physics Institute, Moscow, Russia

org77National Centre for Nuclear Studies, Warsaw, Poland

org78National Institute for Physics and Nuclear Engineering, Bucharest,
Romania

org79National Institute of Science Education and Research, Bhubaneswar, India

org80Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org81Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam,
Netherlands

org82Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org83Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež
u Prahy, Czech Republic

org84Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org85Petersburg Nuclear Physics Institute, Gatchina, Russia

org86Physics Department, Creighton University, Omaha, Nebraska, United States

org87Physics Department, Panjab University, Chandigarh, India

org88Physics Department, University of Athens, Athens, Greece

org89Physics Department, University of Cape Town, Cape Town, South Africa

org90Physics Department, University of Jammu, Jammu, India

org91Physics Department, University of Rajasthan, Jaipur, India

org92Physik Department, Technische Universität München, Munich, Germany

org93Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org94Politecnico di Torino, Turin, Italy

org95Purdue University, West Lafayette, Indiana, United States

org96Pusan National University, Pusan, South Korea

org97Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung, Darmstadt, Germany

org98Rudjer Bošković Institute, Zagreb, Croatia

org99Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org100Russian Research Centre Kurchatov Institute, Moscow, Russia

org101Saha Institute of Nuclear Physics, Kolkata, India

org102School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org103Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org104Sezione INFN, Bari, Italy

org105Sezione INFN, Bologna, Italy

org106Sezione INFN, Cagliari, Italy

org107Sezione INFN, Catania, Italy

org108Sezione INFN, Padova, Italy

org109Sezione INFN, Rome, Italy

org110Sezione INFN, Trieste, Italy

org111Sezione INFN, Turin, Italy

org112SSC IHEP of NRC Kurchatov institute, Protvino, Russia

org113SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France

org114Suranaree University of Technology, Nakhon Ratchasima, Thailand

org115Technical University of Split FESB, Split, Croatia

org116The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org117The University of Texas at Austin, Physics Department, Austin, Texas,
USA

org118Universidad Autónoma de Sinaloa, Culiacán, Mexico

org119Universidade de São Paulo (USP), São Paulo, Brazil

org120Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org121University of Houston, Houston, Texas, United States

org122University of Jyväskylä, Jyväskylä, Finland

org123University of Liverpool, Liverpool, United Kingdom

org124University of Tennessee, Knoxville, Tennessee, United States

org125University of the Witwatersrand, Johannesburg, South Africa

org126University of Tokyo, Tokyo, Japan

org127University of Tsukuba, Tsukuba, Japan

org128University of Zagreb, Zagreb, Croatia

org129Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, France

org130V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia

org131Variable Energy Cyclotron Centre, Kolkata, India

org132Vinča Institute of Nuclear Sciences, Belgrade, Serbia

org133Warsaw University of Technology, Warsaw, Poland

org134Wayne State University, Detroit, Michigan, United States

org135Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary

org136Yale University, New Haven, Connecticut, United States

org137Yonsei University, Seoul, South Korea

org138Zentrum für Technologietransfer und Telekommunikation (ZTT),
Fachhochschule Worms, Worms, Germany