Momentum correlation in pair production by spacetime dependent fields from scattered wave functions

We consider Sauter-Schwinger pair production by electric fields that depend on both time and space, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>E</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>t</a:mi> <a:mo>,</a:mo> <a:mi>z</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>E</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>t</e:mi> <e:mo>,</e:mo> <e:mi>x</e:mi> <e:mo>,</e:mo> <e:mi>y</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> . For space independent fields <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>E</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>t</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> , momentum conservation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>δ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="bold">p</m:mi> <m:mo>+</m:mo> <m:msup> <m:mi mathvariant="bold">p</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy="false">)</m:mo> </m:math> fixes the positron momentum <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi mathvariant="bold">p</s:mi> <s:mo>′</s:mo> </s:msup> </s:math> in terms of the electron momentum <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi mathvariant="bold">p</v:mi> </v:math> . For <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>E</y:mi> <y:mo stretchy="false">(</y:mo> <y:mi>t</y:mi> <y:mo>,</y:mo> <y:mi>z</y:mi> <y:mo stretchy="false">)</y:mo> </y:math> , on the other hand, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:msub> <cb:mi>p</cb:mi> <cb:mi>z</cb:mi> </cb:msub> </cb:math> and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msubsup> <eb:mi>p</eb:mi> <eb:mi>z</eb:mi> <eb:mo>′</eb:mo> </eb:msubsup> </eb:math> are independent. However, previous exact-numerical studies have considered only the probability as a function of a single momentum variable, <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>P</gb:mi> <gb:mo stretchy="false">(</gb:mo> <gb:msub> <gb:mi>p</gb:mi> <gb:mi>z</gb:mi> </gb:msub> <gb:mo stretchy="false">)</gb:mo> </gb:math> , <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>P</kb:mi> <kb:mo stretchy="false">(</kb:mo> <kb:msubsup> <kb:mi>p</kb:mi> <kb:mi>z</kb:mi> <kb:mo>′</kb:mo> </kb:msubsup> <kb:mo stretchy="false">)</kb:mo> </kb:math> or <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>P</ob:mi> <ob:mo stretchy="false">(</ob:mo> <ob:msubsup> <ob:mi>p</ob:mi> <ob:mi>z</ob:mi> <ob:mo>′</ob:mo> </ob:msubsup> <ob:mo>−</ob:mo> <ob:msub> <ob:mi>p</ob:mi> <ob:mi>z</ob:mi> </ob:msub> <ob:mo stretchy="false">)</ob:mo> </ob:math> , but not the correlation <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:mi>P</sb:mi> <sb:mo stretchy="false">(</sb:mo> <sb:msub> <sb:mi>p</sb:mi> <sb:mi>z</sb:mi> </sb:msub> <sb:mo>,</sb:mo> <sb:msubsup> <sb:mi>p</sb:mi> <sb:mi>z</sb:mi> <sb:mo>′</sb:mo> </sb:msubsup> <sb:mo stretchy="false">)</sb:mo> </sb:math> . In this paper, we show how to obtain <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:mi>P</wb:mi> <wb:mo stretchy="false">(</wb:mo> <wb:msub> <wb:mi>p</wb:mi> <wb:mi>z</wb:mi> </wb:msub> <wb:mo>,</wb:mo> <wb:msubsup> <wb:mi>p</wb:mi> <wb:mi>z</wb:mi> <wb:mo>′</wb:mo> </wb:msubsup> <wb:mo stretchy="false">)</wb:mo> </wb:math> by solving the Dirac equation numerically. To do so, we split the wave function into a background and a scattered wave, <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"> <ac:mrow> <ac:mi>ψ</ac:mi> <ac:mo stretchy="false">(</ac:mo> <ac:mi>t</ac:mi> <ac:mo>,</ac:mo> <ac:mi mathvariant="bold">x</ac:mi> <ac:mo stretchy="false">)</ac:mo> <ac:mo>=</ac:mo> <ac:msub> <ac:mrow> <ac:mi>ψ</ac:mi> </ac:mrow> <ac:mrow> <ac:mi>back</ac:mi> <ac:mo>.</ac:mo> </ac:mrow> </ac:msub> <ac:mo stretchy="false">(</ac:mo> <ac:mi>t</ac:mi> <ac:mo>,</ac:mo> <ac:mi mathvariant="bold">x</ac:mi> <ac:mo stretchy="false">)</ac:mo> <ac:mo>+</ac:mo> <ac:msub> <ac:mrow> <ac:mi>ψ</ac:mi> </ac:mrow> <ac:mrow> <ac:mi>scat</ac:mi> <ac:mo>.</ac:mo> </ac:mrow> </ac:msub> <ac:mo stretchy="false">(</ac:mo> <ac:mi>t</ac:mi> <ac:mo>,</ac:mo> <ac:mi mathvariant="bold">x</ac:mi> <ac:mo stretchy="false">)</ac:mo> </ac:mrow> </ac:math> , where <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"> <lc:mrow> <lc:msub> <lc:mrow> <lc:mi>ψ</lc:mi> </lc:mrow> <lc:mrow> <lc:mi>back</lc:mi> <lc:mo>.</lc:mo> </lc:mrow> </lc:msub> <lc:mo>∝</lc:mo> <lc:mi>exp</lc:mi> <lc:mo stretchy="false">(</lc:mo> <lc:mo>±</lc:mo> <lc:mi>i</lc:mi> <lc:mi>p</lc:mi> <lc:mi>x</lc:mi> <lc:mo>+</lc:mo> <lc:mtext>gauge term</lc:mtext> <lc:mo stretchy="false">)</lc:mo> </lc:mrow> </lc:math> . <pc:math xmlns:pc="http://www.w3.org/1998/Math/MathML" display="inline"> <pc:mrow> <pc:msub> <pc:mrow> <pc:mi>ψ</pc:mi> </pc:mrow> <pc:mrow>