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La ge-N kine ic heo y o highly occupied sys ems
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Accep ed e sion (Final d a )
Walz, R.; Bogusla ski, Ki ill; Be ges, J.
Walz, R., Bogusla ski, K., & Be ges, J. (2018). La ge-N kine ic heo y o highly occupied sys ems.
Physical Re iew D, 97(11), A icle 116011. h ps://doi.o g/10.1103/PhysRe D.97.116011
2018
La ge-Nkine ic heo y o highly occupied sys ems
R. Walz,1,* K. Bogusla ski,2,†and J. Be ges1,‡
1Ins i u ü Theo e ische Physik, Uni e si ä Heidelbe g,
Philosophenweg 16, 69120 Heidelbe g, Ge many
2Depa men o Physics, Uni e si y o Jy äskylä, P.O. Box 35, 40014 Uni e si y o Jy äskylä, Finland
(Recei ed 5 Feb ua y 2018; published 12 June 2018)
We conside an e ec i e kine ic desc ip ion o quan um many-body sys ems, which is no based on a
weak-coupling o dilu eness expansion. Ins ead, i employs an expansion in he numbe o ield
componen s No he unde lying scala quan um ield heo y. Ex ending p e ious s udies, we demons a e
ha he la ge-Nkine ic heo y a nex - o-leading o de is able o desc ibe impo an aspec s o highly
occupied sys ems, which a e beyond s anda d pe u ba i e kine ic app oaches. We analyze he unde lying
quasipa icle dynamics by compu ing he e ec i e sca e ing ma ix elemen s analy ically and sol e
nume ically he la ge-Nkine ic equa ion o a highly occupied sys em a om equilib ium. This allows us
o compu e he uni e sal scaling o m o he dis ibu ion unc ion a an in a ed non he mal ixed poin
wi hin a kine ic desc ip ion, and we compa e o exis ing la ice ield heo y simula ion esul s.
DOI: 10.1103/PhysRe D.97.116011
I. INTRODUCTION
A ully mic oscopic desc ip ion o he eal- ime dynam-
ics o quan um many-body sys ems in e ms o quan um
ield heo y can be e y demanding. O en, e ec i e
heo ies wi h a well-de ined ange o alidi y a some
(long) ime and dis ance scales p o ide an e icien al e -
na i e desc ip ion. A well-known example is kine ic heo y,
which desc ibes he s a e o he sys em in e ms o a
classical phase-space dis ibu ion o pa icles, ð ; x;pÞ,a
ime wi h posi ion xand momen um p[1].
Acco dingly, he de i a ion o kine ic heo y om he
unde lying quan um ield heo y in ol es a se ies o c ucial
assump ions [2–6]. An impo an condi ion is ha he de
B oglie wa eleng h ∼1=jpjo ele an (quasi)pa icles
mus be small compa ed o he mean ee pa h be ween
collisions. O he wise, a desc ip ion in e ms o classical
pa icles wi h a well-de ined posi ion and momen um
be ween collisions would no be alid. Likewise, quan um
in e e ence e ec s be ween successi e sca e ing e en s
should no spoil a desc ip ion in e ms o independen
sca e ings. The exis ence o quasipa icle modes wi h a
well-de ined dispe sion ela ion ωðpÞ ansla es in he
language o quan um ield heo y o su icien ly na ow
peaks o he spec al unc ion [7].
These condi ions can be o en me in he p esence o a
su icien ly weak coupling o small dilu eness pa ame e
con olling he s eng h o he sca e ings. In pa icula ,
con olled pe u ba i e kine ic desc ip ions exis o e -
mionic quan um ield heo ies and scala ield heo ies
close o equilib ium whe e he ele an modes wi h
momen a o he o de o he empe a u e ha e occupancies
o o de 1 [2]. Likewise, pe u ba i e desc ip ions exis a
om equilib ium [8–10] i he occupancies o ypical
pa icle modes a e no oo high such ha ≪1=λwi h
λ ep esen ing he ele an coupling cons an o dilu eness
pa ame e . Though gauge heo ies a e mo e in ol ed,
pe u ba i e kine ic desc ip ions dealing wi h he p oblem
o quan um in e e ence ha e been gi en [11–13].
Much less is known abou e ec i e kine ic desc ip ions
o gene al a - om-equilib ium si ua ions. P essing appli-
ca ions conce n sys ems in which he occupancies o
ele an modes a e nonpe u ba i ely la ge ( ∼1=λ) such
ha a pe u ba i e powe coun ing in e ms o a small
coupling pa ame e ails.
An impo an example conce ns he ea ly s ages o a
ela i is ic hea y-ion collision ( o ecen e iews, see
Re s. [14,15] and [11–13,16] o cu en pe u ba i e
kine ic desc ip ions). In his si ua ion, ypical gauge boson
occupancies can become nonpe u ba i ely la ge a low
momen a below he Debye mass scale. These modes may
in luence he e olu ion o impo an quan i ies like he
longi udinal p essu e PLo he expanding plasma, e idence
o which was ound om eal- ime la ice simula ions
[17–19]. Recen s udies ha e ound uni e sal scaling
*[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
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he C ea i e Commons A ibu ion 4.0 In e na ional license.
Fu he dis ibu ion o his wo k mus main ain a ibu ion o
he au ho (s) and he published a icle’s i le, jou nal ci a ion,
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PHYSICAL REVIEW D 97, 116011 (2018)
2470-0010=2018=97(11)=116011(22) 116011-1 Published by he Ame ican Physical Socie y
beha io o in a ed modes [20] ha may be connec ed o
non i ial ield con igu a ions [21]. The possible exis ence
and in luence o an enhanced low-momen um egion o
non-Abelian plasmas ou o equilib ium ha e been ex en-
si ely discussed in he li e a u e [22–29], as ha e me hods
o accessing spec al in o ma ion a he Debye scale and
below [30–33].
Rema kably, longi udinally expanding non-Abelian
plasmas and sel -in e ac ing scala ield heo ies a e ound
o sha e impo an uni e sal aspec s o hei a - om-
equilib ium e olu ion [17,34]. Simila sel -simila scaling
p ope ies a e known o a wide a ie y o highly occupied
sys ems. These include ela i is ic scala sys ems, o en
used in in la iona y models o he ea ly Uni e se a e a
pe iod o esonan pa icle p oduc ion [9,35–37], and
non ela i is ic sys ems such as ul acold quan um gases
o o he condensed ma e sys ems a e a s ong quench
[38–41]. Cha ac e is ic in a ed p ope ies o hese highly
occupied sys ems u n ou o be quan i a i ely he same o
bo h ela i is ic and non ela i is ic models [6,42]. The
uni e sal scaling p ope ies a e associa ed o a non he mal
eno maliza ion g oup ixed poin [37,43] de ining a
uni e sali y class ou o equilib ium, which encompasses
non ela i is ic and ela i is ic N-componen ield heo ies
[6,44], scala sys ems in di e en geome ies [17],in
di e en spa ial dimensions [45,46], and i can e en be
obse ed o a ac i e qua ic in e ac ions as long as mean
in e ac ions a e epulsi e [47]. Pe u ba i e kine ic
app oaches [48–50] b eak down a such la ge ypical
occupa ion numbe s ≳1=λand a e no able o ep oduce
key ea u es o his low-momen um dynamics [6].
To desc ibe he e olu ion also o nonpe u ba i ely la ge
occupa ion numbe s, we conside an e ec i e kine ic
desc ip ion o scala sys ems ha is no based on a
weak-coupling o dilu eness expansion. De eloped in
Re s. [6,45], i exploi s he ac ha o en one desc ibes
complex many-body p oblems wi h mo e han one pa icle
species. In his case, al e na i e kine ic desc ip ions wi h an
ex ended ange o alidi y may be de i ed based on
nonpe u ba i e expansions in he numbe o species
a ailable. Fo scala ield heo ies wi h Nspecies and
qua ic sel -in e ac ions, his esul s om a la ge-Nexpan-
sion o nex - o-leading o de (NLO) based on a wo-pa icle
i educible (2PI) esumma ion o sel -ene gy diag ams
[37,39,43,51,52], which ansla es o a e ex esumma ion
in he kine ic amewo k [6,45].
So a , he la ge-Nkine ic heo y a NLO has been
success ully applied o analy ically compu e he sel -
simila i y exponen s nea non he mal ixed poin s a low
momen a, ag eeing well wi h la ice esul s [6]. Howe e , a
comple e cha ac e iza ion o he nonpe u ba i e in a ed
egime in ol es also he scaling o m o he dis ibu ion
unc ion, which has no been es ablished om he la ge-N
kine ic heo y ye . In his wo k, we p esen he i s
nume ical solu ion o he la ge-Nkine ic equa ion applied
o he uni e sal low-momen um scaling egime in h ee
spa ial dimensions. Ou esul s a e ound o compa e a he
well o a ailable la ice simula ion da a o he unde lying
ield heo y, in pa icula , es ablishing a ∼jpj−4 ail o he
dis ibu ion in he egime domina ed by numbe conse a-
ion. We analyze in de ail he ange o alidi y and
quasipa icle pic u e o he la ge-No “ e ex- esummed”
kine ic heo y and show how i encompasses and ex ends
s anda d pe u ba i e desc ip ions.
The pape is o ganized as ollows. In Sec. II, we conside
scala N-componen ield heo y. S a ing om ela i is ic
models wi h qua ic sel -in e ac ion, we discuss he non-
ela i is ic low-ene gy limi ele an , e.g., also o he
desc ip ion o ul acold Bose gases. Sec ion III summa izes
main aspec s o pe u ba i e kine ic heo y, be o e we
p esen he la ge-Nkine ic desc ip ion in Sec. IV. The
la e has an ex ended ange o alidi y based on he
inclusion o e ex co ec ions, which is analyzed in de ail
in Secs. Vand VI. We p esen a nume ical solu ion o he
la ge-Nkine ic heo y o he desc ip ion o a non he mal
ixed poin in Sec. VII. A e concluding in Sec. VIII,we
end wi h wo Appendixes on calcula ional de ails o he
collision in eg als (Appendix A) and on in eg a ion boun-
da ies (Appendix B).
II. RELATIVISTIC AND NONRELATIVISTIC
SCALAR FIELDS
We conside an OðNÞsymme ic quan um ield heo y
o he ield componen s φað ; xÞ,a¼1;…;N wi h ime
and space a iable xin h ee dimensions and qua ic sel -
in e ac ions. Fo N¼4, he ela i is ic model desc ibes he
Higgs sec o o he S anda d Model o pa icle physics [53].
In he con ex o low-ene gy desc ip ions o quan um
ch omodynamics, such a model encodes he h ee pions
and he sigma esonance. In la on models o ea ly
Uni e se cosmology o en employ ela ed mul icomponen
ield heo ies [54]. In a non ela i is ic se ing, he
Heisenbe g magne o N¼3is a p ominen example,
and he case N¼2can be used o desc ibe he wo eal
componen s o a complex Bose ield in sys ems o ul acold
a omic gases domina ed by s-wa e sca e ing [55].
The conside ed ela i is ic quan um heo y is desc ibed,
on a classical le el, by he ac ion
S½φ¼Z ;x1
2∂μφa∂μφa−m2
2φaφa−λ
4!NðφaφaÞ2ð1Þ
wi h he no a ion R ;x≡Rd Rd3xand he ( eno malized)
mass mand coupling pa ame e λ. He e, summa ion o e
epea ed Lo en z indices μ¼0;…;3and ield indices
a¼1;…;N is implied. We will always employ na u al
uni s, wi h he speed o ligh , Bol zmann’s cons an ,
and he educed Planck cons an equal o uni y,
i.e., c¼kB¼ℏ¼1.
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
116011-2
Fo p ocesses wi h cha ac e is ic momen a below he
mass scale m, one may expec an e ec i ely non ela i is ic
desc ip ion o become ele an e en o he ela i is ic
mic oscopic model (1). Mo e gene ally, he desc ip ions o
ul acold quan um gases o o he condensed ma e sys ems
ypically employ non ela i is ic ield heo ies. One may
ha e in mind he phenomenologically impo an case o an
N¼2-componen non ela i is ic ield heo y, which can
equi alen ly be desc ibed in e ms o a complex ield
ϕð ; xÞ. Following s anda d p ocedu es [56], he e ec i e
low-ene gy desc ip ion may hen be cha ac e ized by he
non ela i is ic ac ion
Sn ½ϕ;ϕ¼Z ;xϕi∂ þ∇2
2mϕ−g
2ðϕϕÞ2:ð2Þ
He e we also in oduced he e ec i e non ela i is ic cou-
pling g, which is no longe dimensionless and may be
ela ed o he ela i is ic pa ame e s as [6]
g∼λ
m2:ð3Þ
Fo dilu e Bose sys ems, gcan be ela ed o he s-wa e
sca e ing leng h, gi en by a¼mg=ð4πÞ[56].
Fo he non ela i is ic quan um heo y in Eq. (2), he
expec a ion alue o he pa icle densi y n¼hϕϕiis
conse ed, and we will conside spa ially homogeneous
sys ems. The densi y nand sca e ing leng h acan be used
o de ine a cha ac e is ic “cohe ence leng h”o which he
in e se is he momen um scale
Q¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
16πan
p∼ffiffiffiffiffiffiffiffiffi
mgn
p:ð4Þ
We also de ine he “dilu eness pa ame e ”
ζ¼ffiffiffiffiffiffiffiffi
na3
p∼Qmg; ð5Þ
which p o ides a dimensionless expansion pa ame e o
he non ela i is ic sys em, simila o he dimensionless
coupling λ o he ela i is ic sys em. Speci ically, wi h (3),
we ob ain ζ∼ðQ=mÞλ.
We emphasize ha o he ela i is ic heo y he pa icle
numbe is no conse ed in gene al. Howe e , o he
highly occupied sys em conside ed in Sec. VII, an app ox-
ima ely conse ed pa icle numbe is dynamically gene -
a ed a a non he mal ixed poin such ha he non ela i is ic
heo y and he ela i is ic one can be in he same uni e -
sali y class o in a ed scaling phenomena [6].
III. PERTURBATIVE KINETIC THEORY
The de i a ion o pe u ba i e kine ic equa ions om he
unde lying quan um-s a is ical ield heo y employs an
expansion in e ms o a small coupling λ≪1o small
dilu eness pa ame e ζ≪1, oge he wi h a g adien
expansion o no - oo-ea ly imes [3,4]. The phase-space
dis ibu ion unc ion o pa icles, ð ; pÞ, desc ibing he
s a e o he spa ially homogeneous sys em a ime and
momen um p, is ob ained om he expec a ion alue o
wo- ield co ela o s e alua ed a equal imes.
Mo e p ecisely, o he ela i is ic ield heo y, he ime
de i a i e o he an icommu a o expec a ion alue
h φa;φbgi≡hφaφbþφbφaide e mines (in he absence
o ex e nal o ces) he change o he dis ibu ion unc ion
acco ding o [45]
Z∞
0
dω
2πω∂
∂ h φa;φbgið ; ω;pÞ≡∂ ð ; pÞ
∂ δab:ð6Þ
He e, he equency ωand spa ial momen um pa ise
om he Fou ie ans o m wi h espec o he ela i e
space- ime a gumen s o he wo ields, while he emaining
ime dependence desc ibes he b eaking o ime- ansla ion
in a iance o he spa ially homogeneous sys em ou o
equilib ium. The kine ic desc ip ion in ol es he
p ojec ion on o posi i e equency con ibu ions by in e-
g a ing o e ω, espec i ely, and we exploi OðNÞsym-
me y assuming no spon aneous symme y b eaking such
ha h φa;φbgi∼δab.
The kine ic equa ion may hen be compu ed pe u ba-
i ely by aking in o accoun in e ac ion e ec s, which a e
subsumed in o he “collision e m”C½  o ob ain
∂ ð ; pÞ
∂ ¼C½ ð ; pÞ:ð7Þ
In i s ange o alidi y, he leading con ibu ions o C½ in a
coupling expansion and an expansion o lowes o de in
g adien s o he massi e scala ield heo y (1) lead o he
well-known Bol zmann equa ion wi h he collision in eg al
o elas ic 2↔2sca e ings [56],
C el½ ð ;pÞ
¼Zl;q;
λ2ðNþ2Þ
6N2I2↔2½ ð ;p;l;q; Þ
×ð2πÞ4δð3Þðpþl−q− Þδðω el
pþω el
l−ω el
q−ω el
Þ
2ω el
p2ω el
l2ω el
q2ω el
;ð8Þ
wi h he no a ion Rq≡Rd3q=ð2πÞ3and he ela i is ic
dispe sion
ω el
p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2þp2
q:ð9Þ
The unc ional I2↔2½ con ains he dis ibu ion unc ions
p≡ ð ; pÞdesc ibing he changes by loss o gain h ough
sca e ing:
LARGE-NKINETIC THEORY FOR HIGHLY OCCUPIED …PHYS. REV. D 97, 116011 (2018)
116011-3
I2↔2½ ð ; p;l;q; Þ
¼ð pþ1Þð lþ1Þ q − p lð qþ1Þð þ1Þ
≈
≫1ð pþ lÞ q − p lð qþ Þ:ð10Þ
In he las equa ion, we gi e he app oxima e exp ession o
la ge occupancies ha will be use ul la e .
Since he o e all collision e m C½ is o o de λ2,
u he pe u ba i e co ec ions o e ms appea ing in he
in eg and o (8) a e subleading. In pa icula , i allows one
o employ in he in eg and a well-de ined dispe sion
ela ion (9). Ph ased in e ms o he unde lying ield heo y,
his leads o a quasipa icle o m o he spec al unc ion
gi en by he expec a ion alue o he commu a o o wo
ields [8,45]:
h½φa;φbiðω;pÞ¼
Oðλ0Þ2πsgnðωÞδðω2−ðω el
pÞ2Þδab:ð11Þ
F om highe o de s in he coupling, he spec al unc ion
would ecei e co ec ions leading o a mass shi and
nonze o wid h o he spec al unc ion encoding “o -shell”
con ibu ions o p ocesses. Howe e , since hey a e o
highe o de in he pe u ba i e powe coun ing o he
collision e m, we do no conside hem he e. Since he e
a e only elas ic collisions con ibu ing o his o de , he
pa icle numbe is a i icially conse ed. Inelas ic p ocesses
can also be aken in o accoun by going o highe o de in
he coupling [44]. S a ing om a gene al ou -o -
equilib ium s a e, such inelas ic p ocesses a e ele an o
desc ibe he app oach o he mal equilib ium a la e imes,
since o he wise a he mal dis ibu ion wi h chemical
po en ial o he pa icle numbe would appea e en in
he absence o a conse ed numbe . Essen ially, neglec ing
inelas ic p ocesses limi s he ime un il which he app oxi-
ma ion can be applied [57]. Fo he pu poses o his sec ion,
going beyond he gi en o de is no necessa y.1
Simila ly, o he non ela i is ic dispe sion ωp¼
jpj2=2ma lowes pe u ba i e o de , he collision e m
o he kine ic equa ion o he heo y wi h ac ion (2)
becomes [10]
Cn ½ ð ; pÞ¼Zl;q;
2g2I2↔2½ ð ; p;l;q; Þð2πÞ4
×δð3Þðpþl−q− Þ
×δðωpþωl−ωq−ω Þ:ð12Þ
The quad a ic dispe sion ela ion a his o de can also be
iewed as a ising om he low-momen um limi o he
abo e ela i is ic collision in eg al (8) [9,50]. We no e ha
aking in o accoun subleading co ec ions o non ela i -
is ic heo ies one gene ally has a Bogoliubo dispe sion
ela ion wi h a quad a ic dispe sion a highe and a linea
dispe sion a lowe momen a i a Bose-Eins ein condensa e
exis s [10], which we do no conside he e.
The pe u ba i e powe coun ing o λ≪1leading o
(8),o (12) o ζ≪1, wi h elas ic 2↔2sca e ings as in
(10) assumes ha he ele an occupancies p o ypical
momen a a e no oo high. Mo e p ecisely, only o
p≪1=λin he ela i is ic, o p≪1=ζin he non-
ela i is ic case, he highe -o de co ec ions a e pa ame i-
cally small. Be o e we discuss his issue in mo e de ail
below in Sec. VI, we will in oduce in he ollowing an
al e na i e kine ic desc ip ion based on a la ge-Nexpansion
o he unde lying quan um ield heo y.
IV. LARGE-NKINETIC THEORY
The s anda d Bol zmann equa ion desc ibed in he las
sec ion is based on a weak-coupling expansion, which
es ic s he ange o alidi y o he kine ic heo y o
pe u ba i e p oblems. Howe e , o he N-componen
ield heo y, an al e na i e kine ic desc ip ion wi h an
ex ended ange o alidi y may be de i ed based on a
nonpe u ba i e expansion in N. Fo de ails abou i s
de i a ion om he unde lying quan um ield heo y, we
e e o Re s. [6,56]. He e, we gi e he ele an exp essions
ha a e used below o sol e he la ge-Nkine ic equa ion.
A la ge N, he classical ac ion (1) scales p opo ional o
N, employing φaφa∼N. Genuine quan um co ec ions due
o sca e ings appea a subleading o de s in a la ge-N
expansion; i.e., hey a e down by ac o s o 1=N compa ed
o classical con ibu ions [39,52]. In pa icula , he spec al
unc ion a leading o de (LO) in a la ge-Nexpansion eads
h½φa;φbiðω;pÞ¼
LOla geN2πsgnðωÞδðω2−ðω el
pÞ2Þδab;ð13Þ
whe e, in con as o he lowes -o de pe u ba i e Eq. (9),
he dispe sion o he ela i is ic heo y now con ains an
e ec i e mass e m M2:
ω el
p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M2þp2
q:ð14Þ
A LO, he e ec i e mass e m is gi en by he gap equa ion
[6,52]
M2¼m2þλ
6Zp
ð 0;pÞ
ω el
pð15Þ
e alua ed a some gi en ime 0. The mass e m is cons an
a lowes o de in he g adien expansion unde lying kine ic
desc ip ions [4].
We will desc ibe in he ollowing ha a NLO in he
la ge-Nexpansion he e is a well-de ined e ec i e kine ic
desc ip ion in e ms o sca e ings be ween quasipa icles.
Simila o he p e ious sec ion, we s a by conside ing he
1Fo ins ance, o a ela i is ic scala ield heo y desc ibing
in e ac ing pions in he con ex o hea y-ion collisions, con-
di ions o a conse ed pa icle numbe densi y and he ime
in e al in which his app oxima ion can be us ed ha e been
discussed in Re s. [58–60].
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
116011-4

ela i is ic heo y and ex end he discussion o he non-
ela i is ic case in he end.
To discuss subleading co ec ions in he 1=N expansion,
i is con enien o employ he auxilia y ield o mula ion o
he same model [52]. Fo his pu pose, we ew i e he
o iginal ac ion (1) by in oducing an auxilia y ield χðxÞas
S½φ;χ¼−Z ;x1
2φað□þm2Þφa−
3N
2λχ2þ1
2χφaφa:
ð16Þ
In eg a ing ou χin he de ining unc ional in eg al yields
he o iginal ac ion, and om he Heisenbe g equa ions o
mo ion, one sees ha he auxilia y ield ep esen s he
composi e ope a o
χðxÞ¼ λ
6NφaðxÞφaðxÞ:ð17Þ
While he auxilia y ield is no a dynamical deg ee o
eedom, i can be used o con enien ly exp ess sca e ing
co ec ions in e ms o he expec a ion alue
Dðx−yÞ≡hχðxÞχðyÞi−hχðxÞihχðyÞi:ð18Þ
Since χ ep esen s a wo-poin unc ion acco ding o (17),
he unc ion Dðx; yÞencodes a ou -poin unc ion o
e ex. Speci ically, a NLO in he 1=N expansion, sca e -
ings a e media ed by (18) [52]. This is indica ed in Fig. 1,in
which dashed lines ep esen he wo-poin unc ion (18) in
Fou ie space. The modi ied e ex a NLO is shown o
sca e ings in he s, , and uchannels, espec i ely.
Acco dingly, he e ec i e kine ic equa ion a NLO is
gi en by he same kine ic equa ion (7), howe e wi h he
di e en collision e m [6]
C el
NLO½ ð ; pÞ¼Zl;q;
λ2
e ð ; p;l;q; Þ
6NI2↔2½ ð ; p;l;q; Þ
×ð2πÞ4δð3Þðpþl−q− Þ
×δðω el
pþω el
l−ω el
q−ω el
Þ
2ω el
p2ω el
l2ω el
q2ω el
:ð19Þ
In he de i a ion o he collision in eg al, he LO exp ession
o he spec al unc ion (13) is used since i s subleading
co ec ions in 1=N would esul in subleading co ec ions
o he collision in eg al, which a e pa o he la ge-N
kine ic heo y a nex - o-nex - o-leading o de (NNLO),
and a e hus omi ed. The ime- and momen um-dependen
e ec i e coupling unc ion
λ2
e ð ;p;l;q; Þ≡λ2
31
j1þΠ el
Rð ;ω el
pþω el
l;pþlÞj2
þ1
j1þΠ el
Rð ;ω el
p−ω el
q;p−qÞj2
þ1
j1þΠ el
Rð ;ω el
p−ω el
;p− Þj2ð20Þ
inco po a es he e ex co ec ions o he di e en sca e -
ing channels acco ding o Fig. 1. The appea ance o he
eno malized one-loop e a ded sel -ene gy
Π el
Rð ;ω;pÞ¼ λ
12 Zq
ð ;p−qÞ
ω el
qω el
p−q1
ω el
qþω el
p−q−ω−iϵ
þ1
ω el
q−ω el
p−q−ω−iϵþ1
ω el
q−ω el
p−qþωþiϵ
þ1
ω el
qþω el
p−qþωþiϵð21Þ
in he denomina o o Eq. (20) is he esul o a geome ic
se ies summa ion o an in ini e numbe o sca e ing
p ocesses a NLO in he la ge-Nexpansion [51].We
emphasize ha Π el
R, and hus also λ2
e , is ime dependen
since i depends on he e ol ing dis ibu ion unc ion.
F om (20), one obse es ha o jΠ el
Rj≪1, which is he
case o weak enough coupling (λ≪1) and no - oo-la ge
ypical occupancies ( ≪1=λ), he e ex co ec ions
encoded in he momen um-dependen e ec i e coupling
become i ele an such ha λ2
e ≃λ2. In his case, he
collision e m (19) essen ially desc ibes s anda d pe u ba-
i e 2↔2sca e ings, howe e a la ge N.2In con as , o
high cha ac e is ic occupancies wi h ∼1=λ, he collision
e m (19) can be s ongly modi ied i Π el
Rs a s o become
o o de 1. We will discuss he co esponding beha io o
he e ec i e coupling in mo e de ail in he ollowing
sec ions, o which we will in oduce he non ela i is ic
e ec i e kine ic equa ion ele an a low momen a below.
Simila o he lowes -o de pe u ba i e kine ic equa ion
o Sec. III, he la ge-Nkine ic heo y a NLO only in ol es
elas ic sca e ing p ocesses. The lack o inelas ic p ocesses
implies conse a ion o he pa icle numbe densi y
n¼Rp ð ; pÞ¼cons , which ollows om he kine ic
equa ion (7) and RpC el
NLO½ ð ; pÞ¼0. Taking in o accoun
inelas ic p ocesses is possible by going beyond NLO,
which is, howe e , beyond he scope o he p esen s udy
ppp
lll
qqq
p+l p-q p-
FIG. 1. Sca e ing p ocesses a nex - o-leading o de in he
la ge-Nexpansion, which a e media ed by an e ec i e in e -
ac ion. While he solid lines ep esen pa icles wi h gi en 4-
momen a, he dashed line ep esen s he unc ion (18) in Fou ie
space desc ibing s-, -, and u-channel exchange.
2The p e ac o ðNþ2Þ=ð6N2Þin Eq. (8) becomes 1=ð6NÞa
NLO in he la ge-Nexpansion.
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ha aims o p o ide a kine ic desc ip ion o numbe -
conse ing dynamics nea non he mal ixed poin s
[6,44,47]. Mo eo e , he e, he condensa e o ma ion ime
di e ges wi h olume ∼V1=αwi h posi i e scaling exponen
αas shown in la ice simula ions [6]. Hence, no eme gence
o a condensa e is expec ed wi hin a ini e ime o he
in ini e olume conside ed, which is consis en wi h he
sel -simila e olu ion ha we will obse e in nume ical
calcula ions o he la ge-Nkine ic heo y in Sec. VII.
The e o e, we will conside he collision in eg al (19)
wi hou including a condensa e in he ollowing. Fu he
discussions on he dis inc ion be ween he pe u ba i e and
nonpe u ba i e egimes can be ound in Re . [61].
To simpli y he ollowing discussion and o make he
connec ion o ul acold a oms, we will es ic ou sel es o
momen a below he (e ec i e) mass scale jpj≪Mand
desc ibe he dynamics in e ms o a non ela i is ic quan um
ield heo y. No e ha also ela i is ic heo ies wi h m¼0bu
M>0may be desc ibed by he non ela i is ic limi o small
momen a. To be consis en wi h he non ela i is ic heo y
de ined by (2), we will use he symbol m o he mass.
Following along he lines o Sec. III, we conside i s he
case N¼2 o illus a e he e ec i e kine ic equa ion o a
non ela i is ic complex scala ield, i.e., wi h wo eal ield
componen s. The case o gene al N hen p oceeds acco d-
ingly [62].
Fo he quad a ic dispe sion ela ion, one ob ains [6]
Cn
NLO½ ð ;pÞ
¼Zl;q;
g2
e ½ ð ;ωp−ωq;p−qÞI2↔2½ ð ;p;l;q; Þ
×ð2πÞ4δð3Þðpþl−q− Þδðωpþωl−ωq−ω Þ:ð22Þ
The e ec i e coupling in he collision in eg al eads
g2
e ½ ð ; ω;PÞ¼ g2
j1þΠRð ; ω;PÞj2;ð23Þ
wi h he one-loop e a ded sel -ene gy
ΠRð ;ω;PÞ¼lim
ϵ→0þgZk
ð ;P−kÞ
×1
ωk−ωP−k−ω−iϵþ1
ωk−ωP−kþωþiϵ
ð24Þ
and he momen um di e ence P¼p−q. As o he
ela i is ic heo y, he collision in eg al (22) educes o
i s pe u ba i e exp ession3 o small jΠRj≪1.
Fo la e use, i is help ul o u he e alua e he
exp essions o Cn
NLO and ΠR. Using magni udes o
momen a p¼jpj, and simila ly o q,k, and P, we ind
o an iso opic sys em (de ails a e gi en in Appendix A)
C½ ð ; pÞ¼ m
32π3pZ∞
0
dqq Zpþq
jp−qj
dP
×g2
e ½ ð ; ωp−ωq;PÞ
×Z∞
max ðP;jp2−q2j
PÞ
duu−ðp2−q2Þ2
u3
×I2↔2½  ; p; 1
2u−p2−q2
u;
q; 1
2uþp2−q2
u ð25Þ
and
ΠRð ; ω;PÞ¼ mg
ð2πÞ2PZ∞
0
dkk ð ; kÞ
× log
ðkþP
2Þ2−m2ω2
P2
ðk−P
2Þ2−m2ω2
P2
þiπZjP2þ2mωj
2P
jP2−2mωj
2P
dkk ð ; kÞ;ð26Þ
whe e we ha e d opped he labels o Cn
NLO o sho en he
no a ion. Acco dingly, he e ec i e kine ic equa ion (7)
depends on ime and he magni ude o he momen um p.
V. BEHAVIOR OF THE LARGE-NRESUMMED
EFFECTIVE VERTEX
To discuss he ex ended ange o alidi y o la ge-N
kine ic heo y, we i s conside he e ec i e coupling g2
e
appea ing in he collision in eg al (25), which is a unc ion
o he di e ence in ene gies o he in- and ou going
pa icles ω¼ωp−ωq¼ðp2−q2Þ=2mand o he magni-
ude o he momen um change P¼jp−qjin a sca e ing
e en . I is bene icial o conside limi ing cases o i s
a gumen s o analyze i s beha io . We dis inguish h ee
ypical collision scena ios. In he i s case, he momen um
o a pa icle wi hin a collision is s ongly decele a ed so ha
p≫qand hus 2mω≈p2≈P2. Simila ly, q≫pleads o
he same e ec i e coupling because g2
e ½ ð ; ω;PÞis
symme ic in ω. To ease he discussion, we will he e o e
use ω≥0in he ollowing. In he second case, he
magni ude o he momen um o he pa icle unde going
he collision s ays a he same o de p∼qwhile i changes
i s di ec ion. The nea ly collinea egime is discussed
sepa a ely and cons i u es he hi d scena io.
The e ec i e coupling in (23) can be calcula ed om he
e a ded sel -ene gy ΠRð ; ω;PÞgi en in (26). Fo he
3The ac o o 2 di e ence be ween he la ge-NNLO ex-
p ession (22) and he pe u ba i e collision in eg al in (12) is due
o an omi ed e m ha is o o de NNLO. No e ha a simila
modi ica ion is ound o he ela i is ic heo y, as commen ed on
in oo no e 2.
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
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dis ibu ion unc ion ð ; kÞ ha en e s he in eg als in ΠR,
we assume ha o l¼0, 1, 2 in eg als o he o m
Zdkkl ð ; kÞ∼Klþ1 ð ; KÞð27Þ
a e domina ed a he possibly ime-dependen momen um
scale K, i i lies wi hin he in eg a ion limi s. This scale K
can be de ined as he momen um in which k2 ð ; kÞis
maximal,
k2 ð ; kÞjk¼K¼max ðk2 ð ; kÞÞ;ð28Þ
such ha i p o ides he dominan con ibu ions o he
pa icle numbe densi y
n∝Zd3k
ð2πÞ3 ð ; kÞ∼K3 ð ; KÞ:ð29Þ
Fo he in eg als in (27) o con e ge be ween he maximal
limi s o 0 and ∞, he dis ibu ion unc ion ð ; kÞshould
all o as e han k−3a la ge momen a k≳Kand dec ease
mo e slowly han k−1a low momen a k≲K, o no
dec ease he e a all.
A. Dispe si e egime, P2≈2mω
We s a wi h he egime p≫q. Then, one has P≈p
and ω≈ωp¼p2=2m≈P2=2m. The second ela ion s a es
ha he ene gy di e ence o in- and ou going momen a ω
ollows he non ela i is ic dispe sion ela ion wi h momen-
um (di e ence) P, and we e e o his egime as
dispe si e. The one-loop e a ded sel -ene gy (26) in his
limi eads
ΠR ; P2
2m;P¼mg
ð2πÞ2PZ∞
0
dkk ð ;kÞlog
kþP
k−P
þiπZP
0
dkk ð ;kÞ:ð30Þ
I s eal pa in ol es an in eg a ion o e all momen a, and
we can use (27) in he limi ing cases o P≳Kand P≲K.
In he i s case, he in eg and is domina ed a low momen a
k, and we can app oxima e logðkþPÞ−log jk−Pj≈
2k=P þOððk=PÞ3Þ. Simila ly, he second case leads o
logðkþPÞ−log jk−Pj≈2P=k þOððP=kÞ3Þ. Hence, he
limi ing exp essions a e
ReΠR ; P2
2m;P
∼
P≳KmgK ð ; KÞK2
P2ð31Þ
ReΠR ; P2
2m;P
∼
P≲KmgK ð ; KÞ:ð32Þ
Fo he imagina y pa , la ge and small ingoing momen a
P≳Kand P≲Klead o he exp essions
ImΠR ; P2
2m;P
∼
P≳KmgK ð ; KÞK
Pð33Þ
ImΠR ; P2
2m;P
∼
P≲KmgP ð ; PÞ:ð34Þ
In (34), we used ha k ð ; kÞshould be a g owing unc ion
a low momen a o be consis en wi h (27).
To ge he co esponding limi ing exp essions o he
e ec i e coupling, we i s assume ha o ypical so
momen a K he occupa ion numbe ð ; KÞis su icien ly
la ge such ha o he conside ed momen a P he 1 in he
denomina o o g2
e in (23) can be neglec ed, and he
e ec i e coupling eads g2
e ≈g2ððReΠRÞ2þðImΠRÞ2Þ−1.
Since P ð ; PÞis limi ed by K ð ; KÞ, he eal pa (32)
domina es a low momen a P≲K. On he o he hand, he
imagina y pa (33) dec eases mo e slowly han he eal pa
a high momen a P≳Kand is hus la ge . Wi h his, he
e ec i e coupling is pa ame ically
g2
e ½  ; P2
2m;P
∼
P≳K1
ðmK ð ; KÞÞ2
P2
K2ð35Þ
g2
e ½  ; P2
2m;P
∼
P≲K1
ðmK ð ; KÞÞ2:ð36Þ
Acco dingly, i is cons an below Kand ollows he powe
law P2beyond K. We no e ha a e en la ge momen a i
becomes cons an ≃g2when he þ1in he denomina o o
i s de ini ion becomes impo an .
As no ed abo e, he egime q≫pleads o he same
exp essions (35),(36), wi h P≈q. The equency a gumen
ge s a minus sign −q2=2m, which does no change he
alues o g2
e because o i s symme y.
B. Momen um-domina ed egime, P2≳2mω
In he momen um-domina ed egime, we conside he
si ua ion in which in- and ou going momen a a e o he
same o de p∼q. Fo u he simpli ica ions, we also
assume ha he equency di e ence is small,
ω¼ωp−ωq≲P2=2m. This occu s o mos o he sca -
e ing angles cos θpq ¼pq=pq, since his assump ion is
equi alen o he condi ion cos θpq ≲minðq=p; p=qÞ. The
emaining case o nea ly collinea collisions cos θpq ∼1
will be discussed in Sec. VC.
In he conside ed egime, he imagina y and eal pa s o
he one-loop e a ded sel -ene gy (26) become
ReΠRð ;ω;PÞ≈mg
ð2πÞ2PZ∞
0
dkk ð ;kÞ2log
2kþP
2k−Pð37Þ
ImΠRð ; ω;PÞ≈mg
4π
P
2  ; P
22mω
P2:ð38Þ
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Expanding he loga i hm o he eal pa and app oxima ing
he in eg al as in (27), one a i es a simila exp essions as
in he dispe si e case
ReΠRð ; ω;PÞ∼
P≳2KmgK ð ; KÞð2KÞ2
P2ð39Þ
ReΠRð ; ω;PÞ∼
P≲2KmgK ð ; KÞ:ð40Þ
A close look on he loga i hm o he o iginal exp ession in
(26) e eals ha i 2mω≲2KP is sa is ied he es ima e o
lowe momen a (40) is e en alid in he collinea egime
whe e 2mωexceeds P2. Hence, he ull ange o alidi y o
(40) is 2K≳P≳2mω=2K, which may only hold i
ð2KÞ2≳2mω. O he wise, o ð2KÞ2≲2mω, no egion
wi h he alue (40) exis s.
To compu e he e ec i e coupling, we again assume
la ge occupa ion numbe s and hus neglec he 1 in he
denomina o in (23), which yields g2
e ≈g2ððReΠRÞ2þ
ðImΠRÞ2Þ−1. Fo bo h small and la ge momen a P, he
eal pa is la ge han he imagina y pa ReΠR≳ImΠR.
This ollows om 2mω≲P2and, o small momen a
P=2≲K, om ðP=2Þ ð ; P=2Þ≲K ð ; KÞ, while o
la ge momen a P=2≳K, i esul s om ðP=2Þ3 ð ; P=2Þ≲
K3 ð ; KÞ, which a e bo h equi emen s o ð ; kÞand
we e o mula ed below Eq. (27). Hence, he e ec i e
coupling pa ame ically ollows
g2
e ½ ð ; ω;PÞ∼
P≳2K1
ðmK ð ; KÞÞ2
P4
ð2KÞ4ð41Þ
g2
e ½ ð ; ω;PÞ∼
P≲2K1
ðmK ð ; KÞÞ2:ð42Þ
The main di e ence om he dispe si e egime is he s eep
powe law P4a la ge momen a. In e es ingly, he ansi ion
be ween small- and la ge-momen um exp essions p oceeds
a he sligh ly la ge scale 2K.
C. Collinea egime, P2≲2mω
The emaining case is when in- and ou going momen a
a e nea ly collinea , cos θpq ∼1, i.e., he case in which
P2≲2mω. The loga i hm appea ing in ReΠRin (26) can be
w i en as
log 
1−P2ð2kþPÞ2
ð2mωÞ2
−log 
1−P2ð2k−PÞ2
ð2mωÞ2
≈−4P4
ð2mωÞ2
2k
P;ð43Þ
whe e we ha e expanded i in he second line. Assuming
ha he in eg al is domina ed a momen a k∼Kas in he
cases abo e, his expansion is jus i ied o la ge momen a
P≳2K, while he condi ion 2mω≳2KP is addi ionally
equi ed a low momen a P≲2K.
Wi h his, we can eadily es ima e he eal and imagina y
pa s o ΠRas
ReΠRð ; ω;PÞ∼−mgK ð ; KÞð2KÞ2
ðPin Þ2ð44Þ
ImΠRð ; ω;PÞ≈mg
4π
Pin
2  ; Pin
2;ð45Þ
whe e we ha e in oduced he in e se momen um
Pin ¼2mω=P. Recall ha o he eal pa he emaining
si ua ion o 2mω≲2KP o low momen a P≲2Kin he
collinea egime has been discussed in Sec. VB, in which i
led o he exp ession (40).
F om his, we can compu e he e ec i e coupling in he
collinea egime. In e es ingly, he eal pa is nega i e, and
1þReΠRin he denomina o o he e ec i e coupling (23)
may become ze o, leading o a esonan inc ease o he
coupling. O he wise, i ð ; KÞis su icien ly la ge, he 1 in
he denomina o o (23) can again be neglec ed, and he eal
and imagina y pa s o ΠRcan be compa ed in o de o
es ima e g2
e ≈g2ððReΠRÞ2þðImΠRÞ2Þ−1. A i s , we
conside Pin ≳2K. This condi ion ansla es o 2mω≳
2KP and co esponds o he eal pa as gi en by (44).
Using ðPin Þ3 ð ; Pin Þ≲K3 ð ; KÞ o la ge momen a
Pin , one inds ImΠR≲jReΠRj. Simila ly, he case Pin ≲
2K ansla es o 2mω≲2KP, and he eal pa is hen gi en
by (40). Wi h Pin ð ; Pin Þ≲K ð ; KÞ o low momen a,
one again inds ImΠR≲jReΠRj. Thus, as in he momen-
um-domina ed egime, he eal pa domina es he e ec i e
coupling. A low momen a P≲2K, i leads o (42) o
2mω≲2KP, while in all o he cases, in he collinea
egime, one has
g2
e ½ ð ; ω;PÞ∼
1
ðmK ð ; KÞÞ2ð2mωÞ4
ð2KPÞ4:ð46Þ
Hence, di e en om he momen um-domina ed egime,
he e ec i e coupling dec eases he e as P−4.
D. Compa ison o nume ical esul s
We now compu e he e ec i e coupling g2
e nume ically
as de ined in (23). Fo he dis ibu ion unc ion,
we use
ð ; pÞ≃
1
g
A
ðp=BÞκ<þðp=BÞκ>;ð47Þ
which has been sugges ed in Re . [6] o app oxima e
he dis ibu ion unc ion a low momen a du ing he
sel -simila egime. The pa ame e Bis ela ed o he
momen um scale Kde ined in Eq. (28) ia
K¼Bðð2−κ<Þ=ðκ>−2ÞÞ1=ðκ>−κ<Þ, such ha bo h a e o
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
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momen um KSand he ampli ude SðKSÞob ained he e
wi hin la ge-Nkine ic heo y o he o e line no a ion, as
KS¼ −β
e KS≈0.0577 ð76Þ
and SðKSÞ¼ α
e SðKSÞ, and compa e hem o he co e-
sponding quan i ies in Re . [6]. Since in ou p esc ip ion o
he la ge-Nkine ic heo y o e y high occupa ion
numbe s (see oo no e 9) he ampli ude d ops ou o he
kine ic equa ion, i can hus be adjus ed a bi a ily e en
a e he simula ion, depending on he pa icle numbe
densi y in he sys em n∼ SðKSÞK3
S¼ SðKSÞKS
3. Hence,
i is no sui able o a compa ison wi h he la ice esul s. On
he o he hand, being independen o n, he scale KS(o KS)
is ixed by he mass pa ame e m(o by bo h mand he
e e ence ime e ) and enables a quan i a i e compa ison
be ween la ge-Nand la ice simula ion esul s. In Fig. 3 o
Re . [6], he ansi ion scale KSis loca ed wi hin he
momen um ange 0.05 ≲KS≲0.08. This is consis en
wi h ou esul om he la ge-Nkine ic heo y (76).
We no e ha he small de ia ions be ween la ge-N
kine ic and la ice esul s may ha e di e en easons.
Fi s o all, we use he la ge-Nkine ic heo y a NLO
and omi highe o de s in 1=N. Mo eo e , he unc ional
o m measu ed on he la ice in Re . [6] may su e om
ini e- ime e ec s and may sligh ly change a la e imes
beyond he simula ion imes shown he e.10 And inally,
ega ding he discussion below Eq. (67), he obse ed
powe law on he la ice could be a supe posi ion o powe
laws wi h di e en o igins. The e o e, i was o g ea
impo ance o pin down he powe -law exponen 4 in
Eq. (74) ha can be associa ed wi h he la ge-Nkine ic
heo y11 o be able o dis inguish i om o he possible
con ibu ions.
We ha e seen ha he la ge-Nkine ic heo y p o ides an
e en quan i a i ely good desc ip ion o classical-s a is ical
la ice da a. This con i ms i s applicabili y o sys ems wi h
e y high occupa ion numbe s, ex ending pe u ba i e
kine ic amewo ks. The scaling unc ion in Fig. 6and
i s p ope ies a e he main esul s o his sec ion.
C. Nume ical se up
He e, we discuss he nume ical se up ha led o he
scaling solu ion in Fig. 6. To sol e he ixed-poin equa-
ion (73), we s a again wi h he ull kine ic equa ion in (7)
wi h he collision in eg al C½ as gi en by (25).Ou
s a egy is o escale he kine ic equa ion such ha i elaxes
o he ixed-poin equa ion wi h ime. Wi h his, we ollow
Re . [26] in which his s a egy was used o he sel -simila
egion a ha d momen a in non-Abelian gauge heo y.
The e o e, ins ead o using a sel -simila i y ansa z, we
escale he dis ibu ion unc ion and momen a acco ding o
ð ; pÞ≡ α˜
ð ; βpÞ≡ α˜
ð ; ˜
pÞ;ð77Þ
wi h he alues o he scaling exponen s om (66).
Compa ing (77) o he sel -simila e olu ion (65), one
inds ha he scaling unc ion is he s a iona y limi o his
escaled dis ibu ion
˜
ð ; ˜
pÞ→ Sð˜
pÞ.
Plugging (77) in o he kine ic equa ion (7), one a i es a
he escaled kine ic equa ion
∂
˜
ð ; ˜
pÞ
∂log ¼−α
˜
ð ; ˜
pÞþβ˜
p∂
˜
ð ; ˜
pÞ
∂˜
p−C½
˜
ð ; ˜
pÞ;ð78Þ
since he explici ime ac o s cancel. No e ha Eq. (78) is
equi alen o he o iginal kine ic equa ion (7) bu educes o
he ixed-poin o m (73) when
˜
becomes ime indepen-
den . Hence, a s a iona y solu ion o his equa ion co e-
sponds o he scaling unc ion Sð˜
pÞ, as has been no ed
abo e. In his sense, and because o he loga i hmic ime
de i a i e ∂
˜
=∂log , i can be ega ded as a elaxa ion
algo i hm o Sð˜
pÞin ime.
Mo eo e , he o e all ampli ude o
˜
d ops ou o he
kine ic equa ion (78) because we neglec he 1 in he
denomina o o he e ec i e coupling (23) (see also oo -
no e 9). This co esponds o he high-occupancy limi
˜
ð ;
˜
KÞ→∞wi h
˜
ð ; ˜
pÞ=
˜
ð ;
˜
KÞkep ixed o each
momen um. He e,
˜
Kis he ypical ( escaled) in a ed
momen um scale in which pa icle numbe densi y nis
domina ed wi h espec o he dis ibu ion
˜
. Simila ly, he
coupling cons an galso d ops ou o he kine ic equa ion.
To nume ically sol e (78), we disc e ize ime loga i hmi-
cally wi h cons an Δlog ¼log kþ1−log k¼cons
be ween successi e imes kþ1and k. The ime can hen
be calcula ed as k¼ekΔlog . Mo eo e , we choose a
loga i hmically spaced momen um g id o he dis ibu ion
unc ion
˜
ð ; ˜
pÞ, i s de i a i e, and he collision in eg al, in
o de o esol e he dis ibu ion unc ion a e y low
momen a. The g id in ol es Npmomen a be ween ΛIR
and ΛUV such ha he a io be ween successi e momen a is
cons an ˜
pkþ1=˜
pk¼cons . We employed Δlog ¼0.1,
ΛIR ¼0.017,ΛUV ¼17–52, and Np¼100 o he plo s
o his sec ion.
Fo he compu a ion o he collision in eg al and o he
in e pola ion o he dis ibu ion unc ion, we use me hods12
om he GNU Scien i ic Lib a y [71]. In e pola ion is
equi ed since he in eg a ion me hods need con inuous
10Simila ini e- ime a i ac s ha e been obse ed o non-
Abelian gauge heo y, in which a kine ic desc ip ion compa ed
well wi h la ice esul s bu showed sligh di e ences a la e imes
beyond he ime o la ice simula ions when being close o he
non he mal ixed poin [26].
11See also Re . [61] o an al e na i e app oach o his p oblem
and an ou come consis en wi h ou esul s.
12F om Re . [71], we equen ly use he in eg a ion me hod
GSL
_
INTEGRATION
_
CQUAD
, which is pa icula ly sui able o
singula in eg ands. Such occu , o ins ance, in he eal pa
o he one-loop e a ded sel -ene gy ΠR. The employed in e -
pola ion ype is
GSL
_
INTERP
_
AKIMA
.
LARGE-NKINETIC THEORY FOR HIGHLY OCCUPIED …PHYS. REV. D 97, 116011 (2018)
116011-15

unc ions o he in eg and, and i is pe o med based on he
sampling poin s ð˜
pk;
˜
n;k ≡
˜
ð n;˜
pkÞÞ a ime n.Fo
momen a ou side he momen um g id ½ΛIR;ΛUV,wese
he dis ibu ion unc ion o ze o. The e o e, some e ms in
he unc ional I2↔2½
˜
in (10) become ze o when one o he
momen a is ou side o he momen um in e al, and he
collision in eg al loses i s gain-minus-loss s uc u e. To
p e en his, we addi ionally se he whole unc ional
I2↔2½
˜
 o ze o in such cases o educe de ia ions om
pa icle numbe and ene gy densi y conse a ion. This leads
o simpli ica ions o in eg a ion bounda ies wi hin he colli-
sion in eg al (25), which a e u he discussed in Appendix B.
In he nume ical algo i hm, we i s ini ialize he dis-
ibu ion unc ion
˜
0and i s momen um de i a i e
˜
0
0a he
g id poin s ˜
pka ini ial ime. The ime s ep n→ nþ1
ollows he explici Eule me hod
˜
nþ1;k ¼
˜
n;k −Δlog ½α
˜
n;k þβ˜
pk
˜
0
n;k −C½
˜
n:ð79Þ
Since he e alua ion o he collision in eg al a a single
momen um poin pknei he depends on no a ec s he
e alua ion a o he momen a, we pa allelize he pa o ou
sol e in which he collision in eg al is compu ed o each
momen um on he g id.
The accu acy o ou solu ion algo i hm is mainly limi ed
by he in e pola ion o he de i a i e o he dis ibu ion
unc ion
˜
0. Fo a ypical unc ional o m as in Eq. (47), he
ela i e accu acy o ou disc e iza ion was up o 10−2as
compa ed o he analy ical exp ession o he de i a i e.
Al hough inc easing Npmay imp o e he esolu ion, he
compu a ional cos s will g ow, and we hus ound a
comp omise ha s ill p o ided su icien ly accu a e esul s.
D. De ails on he compu a ion o
he scaling unc ion
In Fig. 7, we show he elaxa ion dynamics o he
escaled dis ibu ion
˜
compu ed by he algo i hm in o-
duced abo e. We s a close o he s a iona y o m by
choosing
˜
0as in Eq. (47) wi h κ<¼0and κ>¼3.9.One
obse es ha
˜
quickly app oaches a s a iona y o m,
which can be unde s ood as he scaling unc ion S.
Cu es a imes ≥1.6a e al eady almos ime indepen-
den . The e o e, Sshown in Fig. 6is compu ed as he
a e age o e hese cu es, while he e o ba s a e es i-
ma ed by he s anda d de ia ion in his p ocedu e.
The unc ional o m o Sis ba ely dis inguishable om
i s s a ing o m (47) in Fig. 7; howe e , small de ia ions
a ound he scale KSexis . A low momen a ˜
p≲KSand a
high momen a ˜
p≳KS, he scaling unc ion ollows powe
laws ˜
p−κ<and ˜
p−κ>. We ha e measu ed he spec al
exponen s κiby employing powe -law i s o he espec i e
egions in he scaling unc ion,
κ<¼00.01ðsysÞ
κ>¼3.95 0.05ðsysÞ:ð80Þ
S a is ical e o s a e much smalle han sys ema ic e o s,
which we e es ima ed o con ain possible in a ed and
ul a iole cu o a i ac s.13
To check he s abili y o hese alues, we s a wi h
sligh ly la ge exponen s κ<¼0.5and κ>¼4.5 o he
ini ial dis ibu ion
˜
0wi h he unc ional o m (47). The ime
e olu ion o he exponen s is shown in Fig. 8, whe e we use
he e o es ima es o (80). Indeed, one obse es ha he
exponen s app oach he alues (80) o he scaling unc ion.
VIII. CONCLUSION
In his wo k, we ha e shown ha he la ge-Nkine ic
heo y a NLO can be applied o highly occupied scala
quan um ield heo y, which canno be desc ibed by a
pe u ba i e kine ic amewo k. On he o he hand, o
su icien ly low occupancies o a la ge momen a, i
e ec i ely educes o a pe u ba i e kine ic heo y a la ge
N. Hence, he la ge-Nkine ic heo y ex ends pe u ba i e
desc ip ions and cons i u es a e sa ile ool o s udy he
dynamics o sys ems ou o equilib ium in e ms o
quasipa icles.
An essen ial ing edien o he quasipa icle pic u e, o
ee mo emen be ween collisions and o he p ope
inclusion o quan um in e e ence e ec s e en a high
occupancies, is he e ex esumma ion ha appea s a
NLO in he la ge-Nexpansion [6,39,45,51,52]. We ana-
lyzed he s uc u e o he e ec i e e ex in de ail, which
enabled us o show ha he mean ee pa h Lo quasi-
pa icles a ypical momen um modes is la ge han hei de
B oglie wa eleng h. Mo eo e , he spec al unc ion can be
FIG. 8. The exponen s o app oxima e powe laws ˜
p−κia low
(κ<) and high (κ>) momen a o he escaled dis ibu ion
˜
as
unc ions o ime. The dis ibu ion has been ini ialized as in (47)
wi h A¼27000,B¼1.135,κ<¼0.5, and κ>¼4.5.
13We no e ha a low and high momen a close o he cu o s he
scaling unc ion s a s o show de ia ions om powe -law o
cons an beha io , which is he main sou ce o e o in he powe -
law i s. Inc easing he momen um g id o lowe and la ge
momen a educes he e ec s o hese nume ical a i ac s bu
comes a he p ice o inc eased nume ical cos s.
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
116011-16
app oxima ed by a quasipa icle o m a NLO o he la ge-
N heo y since a peak wid h is supp essed by 1=N. These
a gumen s lead o a well-de ined dispe sion ela ion and
hus a consis en quasipa icle pic u e.
In a second s ep, we applied he la ge-Nkine ic heo y o
he highly occupied egion o scala sys ems a low
momen a cha ac e ized by a uni e sal sel -simila e olu ion
[6,44]. Su passing o me analy ical es ima es o he
scaling exponen s αand βo he sel -simila e olu ion
[6], we sol ed he e ec i e kine ic equa ion nume ically o
he i s ime. The scaling unc ion ob ained, SðjpjÞ, ag ees
well wi h o me la ice simula ion esul s, which is a
s iking con i ma ion o he applicabili y o he la ge-N
kine ic heo y o highly occupied sys ems. I e eals a
powe -law beha io ∼jpj−4a momen a highe han he
ypical momen um KS ha domina es pa icle numbe
densi y and becomes cons an a low momen a.
No explici assump ion on he coupling s eng h has
en e ed he de i a ion o he la ge-Nkine ic heo y.
The e o e, in p inciple, one could use he la ge-Nkine ic
heo y also a mode a e couplings o ini e N. Indeed, i was
shown using 2PI equa ions o NLO in a 1=N expansion
[39,51,52] ha he obse ed sel -simila egime a low
momen a su i es o mode a e alues λ¼1[68].
Howe e , i was a gued wi hin he 2PI amewo k ha o
a la ge coupling λ¼10 inelas ic p ocesses may play an
impo an ole a all imes in he e olu ion [72].Thela ge-N
kine ic heo y in i s p esen o m a NLO, howe e , lacks
such p ocesses.
To be able o simula e a comple e he maliza ion p ocess
wi hin la ge-Nkine ic heo y, s a ing a om equilib ium
and he e olu ion owa d a non he mal ixed poin includ-
ing he subsequen inal he maliza ion dynamics, one
would ha e o go beyond his o de o cap u e inelas ic
p ocesses. While going o NNLO is challenging, he
desc ip ion may be pa ially simpli ied a la e imes ele an
o he inal app oach o he mal equilib ium since he
ypical occupancies become smalle such ha s anda d
pe u ba i e app oxima ions become a ailable again a leas
o small enough couplings.
The la ge-Nkine ic heo y is an example o a kine ic
heo y applicable o highly occupied sys ems, o which
con en ional kine ic app oaches ail. Fo non-Abelian
plasmas, mul iple s udies indica e s ong ields and non-
i ial dynamics a low-momen um modes [17–21,29].An
e ec i e desc ip ion he eo could be an impo an ex en-
sion o kine ic app oaches [7,11–13,16] o he e olu ion o
weakly coupled non-Abelian plasmas and he he mal-
iza ion p ocess in ul a ela i is ic hea y-ion collisions a
high ene gies.
ACKNOWLEDGMENTS
We hank J. P. Blaizo , I. Chan esana, T. Gasenze ,
A. Ku kela, T. Lappi, A. Piñei o O ioli, S. Schlich ing,
and R. Venugopalan o use ul discussions and
collabo a ions on ela ed wo k. K. B. g a e ully acknowl-
edges suppo by he Eu opean Resea ch Council unde
G an No. ERC-2015-COG-681707. This wo k is pa o
and suppo ed by he DFG Collabo a i e Resea ch Cen e
“SFB 1225 (ISOQUANT).”
APPENDIX A: TOWARD SOLVING
THE LARGE-NKINETIC EQUATION
In his Appendix, we sol e some o he in eg als
appea ing in he collision in eg al o he non ela i is ic
la ge-Nkine ic heo y (22) analy ically o d¼3spa ial
dimensions.
1. Collision in eg al
We s a by a e aging he Bol zmann equa ion (7) o e he
solid angle o p. Because o he iso opy o he dis ibu ion
unc ion, he le -hand side does no change, while he igh -
hand side becomes
C½ ð ; pÞ¼ZdΩp
4πC½ ð ; pÞ
¼π
ð2πÞ10 Z∞
0
dll2dqq2d 2ZdΓCðp; l; q; Þ
×δðωpþωl−ωq−ω Þ
×g2
e ½ ð ; ωp−ωq;PÞ
×I2↔2½ ð ; p; l; q; Þ:ðA1Þ
We ha e in oduced he momen um di e ence P≡p−q
o he e ec i e coupling g2
e . In Appendix A2,we
will show ha i only depends on he magni ude
P¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2þq2−2pq cosðθp;qÞ
qand hence on he magni-
udes o he momen a p¼jpjand q¼jqjand on he angle
θp;qbe ween hem. The h ee-dimensional momen um
in eg als in (A1) ha e been spli in o adial and solid
angle pa s acco ding o Rd3q¼R∞
0dqq2RdΩqwi h
RdΩq¼R2π
0dφqR1
−1dcosðθqÞ. All angula in eg als ha e
been included in
ZdΓCðp; l; q; Þ≡ZdΩpdΩldΩqdΩ ð2πÞ3
×δð3Þðpþl−q− Þ:ðA2Þ
Excep o θp;q, he e is no angula dependence in he
esidual e ms o he collision in eg al. In he ollowing, he
espec i e in eg als a e pe o med analy ically.
We s a by using he in eg al ep esen a ion o he del a
unc ion
ð2πÞ3δð3Þðpþl−q− Þ¼Zd3xeiðpþl−q− Þx:ðA3Þ
LARGE-NKINETIC THEORY FOR HIGHLY OCCUPIED …PHYS. REV. D 97, 116011 (2018)
116011-17
Exploi ing RdΩpdΩq¼RdΩqdΩp;q, whe e RdΩp;q
deno es he angula in eg a ion o pa ound he axis in q
di ec ion, and pe o ming in eg a ions o e he angles
excep o θp;q, we a i e a
ZdΓCðp;l;q; Þ¼24ð2πÞ5Z1
−1
dcosðθp;qÞZ∞
0
dxx2
×sinðPxÞ
Px
sinðlxÞ
lx
sinð xÞ
x ;ðA4Þ
whe e we employed R1
−1dyeixy ¼2sinðxÞ=x.Weha e
used ha he collision in eg al does no depend on he pola
angle θqo q, e en a e in eg a ing o e θp;q, such ha he
in eg a ion R1
−1d cosðθqÞ¼2can be pe o med in he end,
which leads o Eq. (A4). Wi h
sinðaÞsinðbÞsinðcÞ¼1
4ð−sinða−b−cÞþsinðaþb−cÞ
þsinða−bþcÞ−sinðaþbþcÞÞ
ðA5Þ
and
Z∞
0
dxsinðaxÞ
x¼π
2sgnðaÞ;ðA6Þ
we ob ain
ZdΓCp; l; q; Þ¼ð2πÞ6
l Z1
−1
d cosðθp;qÞ
P
×ðsgnðPþl− ÞþsgnðP−lþ Þ
−sgnðP−l− Þ−sgnðPþlþ ÞÞ:
ðA7Þ
Taking he in eg als o e land o he collision in eg al
(A1) in o accoun , he sign unc ions in exp ession (A7) can
be con enien ly e alua ed ia
Z∞
0
dlZ∞
0
d ðsgnðPþl− ÞþsgnðP−lþ Þ
−sgnðP−l− Þ−sgnðPþlþ ÞÞ
¼2Z∞
P
d ZPþ
0
dlþZP
0
d ZPþ
P−
dl−Z∞
0
dlZ∞
Pþl
d 
≡2ZΔðPÞ
dld ¼Z∞
P
duZP
−P
d : ðA8Þ
A change o a iables om land o u¼ þland ¼
−lhas been pe o med in he las s ep o Eq. (A8),
abso bing a ac o o 2. Figu e 9 isualizes he egion
o in eg a ion ΔðPÞ. A second ans o ma ion om
d cosðθp;qÞ o dPwi h d cosðθp;qÞ¼−ðP=pqÞdPleads o
C½ ð ; pÞ¼ 1
64π3pZ∞
0
dqZpþq
jp−qj
dPZ∞
P
duZP
−P
d
×qðu2− 2Þδðωpþωðu− Þ=2−ωq−ωðuþ Þ=2Þ
×g2
e ½ ð ; ωp−ωq;PÞ
×I2↔2½  ; p; u−
2;q;uþ
2:ðA9Þ
The nex s ep is he e alua ion o he ene gy-conse ing
del a unc ion. Wi h he quad a ic dispe sion ela ion
ωp¼p2=2m, he del a unc ion in he collision in eg al
(A9) eads
δ1
2mp2þ1
4ðu− Þ2−q2−
1
4ðuþ Þ2
¼2mδðp2−q2−u Þ:ðA10Þ
We use he in eg al o e alua e he del a unc ion
acco ding o
2mZP
−P
d Rð Þδðp2−q2−u Þ
¼2m
uRp2−q2
uΘP−jp2−q2j
u;ðA11Þ
whe e R ep esen s e e y pa o he collision
in eg al depending on . The s ep unc ion in (A11) can
be used o change he in eg a ion bounda ies o u o
R∞
max ðP;jp2−q2j=PÞdu. The esul ing o m o he collision
in eg al is gi en by Eq. (25).
2. Re a ded sel -ene gy
To also simpli y he compu a ion o he e ec i e
coupling g2
e o (23), we pe o m he angula in eg a ions
o he one-loop e a ded sel -ene gy ΠRin (24) analy ically.
FIG. 9. The illed egion ΔðPÞ ep esen s he a ea o in eg a ion
in (A8). The igu e is aken wi h adap ion om Re . [6].
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
116011-18
We i s change he in eg a ion a iable o P−k↦k.
In oducing y≡cosðθP;kÞ, whe e θP;kis he pola angle
be ween he momen a Pand k, one a i es a
ΠRð ; ω;PÞ¼lim
ϵ→0þ
g
ð2πÞ2Z∞
0
dkk2 ð ; kÞZ1
−1
dy
×1
ðP2−2PkyÞ=2m−ω−iϵ
þ1
ðP2−2PkyÞ=2mþωþiϵ:ðA12Þ
No e ha , due o he iso opy o ð ; kÞ,ΠRonly depends
on he absolu e alue o he momen um P¼jPj. Mo eo e ,
i anishes o P¼0since he in eg and becomes iden i-
cally ze o. In addi ion, one obse es ha changing he
equency om ω o −ωco esponds o complex con-
juga ion o he whole exp ession. As a consequence, he
one-loop e a ded sel -ene gy ΠRis eal o ω¼0.
Rew i ing (A12) as
ΠRð ; ω;PÞ¼lim
˜
ϵ→0þ
−mg
ð2πÞ2PZ∞
0
dkk ð ; kÞZ1
−1
dy
×1
y−P2−2mω
2Pk þi˜
ϵþ1
y−P2þ2mω
2Pk −i˜
ϵ;
ðA13Þ
whe e we ha e subs i u ed ˜
ϵ≡mϵ=Pk, enables us o use
he p incipal alue in eg al
lim
ϵ→0þZ1
−1
1
xþDiϵdx¼PV Z1
−1
1
xþDdx
∓iπZ1
−1
δðxþDÞdx
¼log
1þD
−1þD
∓iπΘð1−jDjÞ
ðA14Þ
o each ac ion in Eq. (A13). This leads o
ΠRð ; ω;PÞ¼ −mg
ð2πÞ2PZ∞
0
dkk ð ; kÞΓΠðω;P;kÞðA15Þ
wi h he ke nel
ΓΠðω;P;kÞ¼log
ðP2−2PkÞ2−4m2ω2
ðP2þ2PkÞ2−4m2ω2
−iπΘ1−jP2−2mωj
2Pk 
−Θ1−jP2þ2mωj
2Pk :ðA16Þ
Fo a nume ical ea men , i is use ul o know he
singula poin s o he eal pa o he in eg and
ReΓΠðω;P;kÞand he egion whe e he imagina y pa
ImΓΠðω;P;kÞdoes no anish. The singula i ies o he eal
pa a e gi en by
ksing ¼P
2mjωj
P;ðA17Þ
whe e he case P¼0is excluded since he whole in eg and
is ze o hen. The eal pa can be ew i en as
ReΓΠðω;P;kÞ¼log
ðk−P
2Þ2−m2ω2
P2
ðkþP
2Þ2−m2ω2
P2
:ðA18Þ
Fo he imagina y pa ImΓΠðω;P;kÞo he in eg and
(A16), we will assume ω>0since he case o a nega i e
equency is ela ed o he posi i e equency case by
complex conjuga ion as no ed abo e. Then, ImΓΠðω;P;kÞ
is ze o un il he in eg a ion a iable khas inc eased
su icien ly o ul ill he condi ion k≥jP2−2mωj=2Po
he i s Hea iside s ep unc ion bu is s ill oo small o
ul ill he condi ion o he second s ep unc ion. A e
exceeding jP2þ2mωj=2P, which makes he second s ep
unc ion one as well, he whole exp ession anishes again.
Al oge he , we ind
−
1
πImΓΠðω;P;kÞ¼1i jP2−2mωj
2P≤k≤jP2þ2mωj
2P
0else :
ðA19Þ
Including also he case o nega i e equency, we a i e a
he inal o m o he one-loop e a ded sel -ene gy
in Eq. (26).
APPENDIX B: INTEGRATION BOUNDARIES
As explained in Sec. VII C, he collision in eg al
C½ ð ; pÞas well as he dis ibu ion unc ion ð ; pÞa e
disc e ized on a g id be ween he momen a ΛIR and ΛUV in
ou nume ical app oach. Ou side his domain, we se
ð ; pÞ¼0. Making use o his, he in eg a ion bounda ies
o he collision in eg al (25) can be u he cons ained.
In eg als wi hin he eal and imagina y pa s o he one-
loop e a ded sel -ene gy (26) a e simpli ied o
Re ΠR∶Z∞
0
dk→ZΛUV
ΛIR
dk; ðB1Þ
Im ΠR∶ZjP2þ2mωj
2P
jP2−2mωj
2P
dk→Zmin ðjP2þ2mωj
2P;ΛUVÞ
max ðjP2−2mωj
2P;ΛIRÞ
dk: ðB2Þ
The gain-minus-loss pa
LARGE-NKINETIC THEORY FOR HIGHLY OCCUPIED …PHYS. REV. D 97, 116011 (2018)
116011-19
I2↔2½  ; p; 1
2u−p2−q2
u;q;1
2uþp2−q2
u
ðB3Þ
o he in eg and o he collision in eg al yields non anish-
ing con ibu ions i
ΛIR ≤p≤ΛUV;
ΛIR ≤q≤ΛUV;
ΛIR ≤
1
2ðup2−q2
uÞ≤ΛUV:ðB4Þ
To p ese e he gain-minus-loss symme y, we se he
whole I2↔2 o ze o i one o he condi ions (B4) is no
ul illed, as explained in Sec. VII C. The i s cons ain is
au oma ically ul illed since he momen um pis ex e nally
se o he co ec momen um ange. The qin eg a ion is
changed o
Z∞
0
dq→ZΛUV
ΛIR
dq: ðB5Þ
The ela ion
ΛIR ≤
1
2u2mjωj
u≤ΛUV ðB6Þ
wi h 2mjωj≡jp2−q2jis chosen o be ul illed o plus
and minus signs simul aneously. I can be sol ed o u o
ob ain new in eg a ion bounda ies acco ding o he ollow-
ing p ocedu e.
In a i s s ep, we conside he le inequali y o (B6), i.e.,
ðu2mjωj=uÞ=2≥ΛIR. Using ha u>0, he inequali y
can be ans o med o
ðu−ΛIRÞ2≥Λ2
IR ∓2mjωj;ðB7Þ
whe e he wo cases o minus and plus signs on he igh -
hand side a e dis inguished.
Fo he case o he minus sign, no es ic ion o uis
ob ained i Λ2
IR ≤2mjωj. O he wise, one ob ains
u≤ΛIR −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
IR −2mjωj
qi u<ΛIR;ðB8Þ
u≥ΛIR þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
IR −2mjωj
qi u>ΛIR:ðB9Þ
In case o he plus sign on he igh -hand side o (B7), one
inds he cons ain s
u≥ΛIR þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
IR þ2mjωj
qi u>ΛIR;ðB10Þ
u≤ΛIR −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
IR þ2mjωj
qi u<ΛIR:ðB11Þ
The la e condi ion (B11) excludes u<ΛIR since ualways
has o be posi i e.
In a second s ep, we conside he inequali y on he igh -
hand side o (B6), which can be ans o med o
ju−ΛUVj≤ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV ∓2mjωj
q:ðB12Þ
This leads o he ou condi ions
u≥ΛUV −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV −2mjωj
qi u<ΛUV;ðB13Þ
u≤ΛUV þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV −2mjωj
qi u>ΛUV;ðB14Þ
u≤ΛUV þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV þ2mjωj
qi u>ΛUV;ðB15Þ
u≥ΛUV −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV þ2mjωj
qi u<ΛUV;ðB16Þ
whe e he condi ion (B16) is no cons ain , since
ΛUV −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV þ2mjωj
p≤0.
All conside ed cases and he successi e cons ain s a e
employed a he same ime. This allows us o change he
in eg a ion ange o u o
Z∞
maxðP;2mjωj
PÞ
du
→ZΛUVþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV−2mjωj
p
maxðP;2mjωj
P;ΛIRþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
IRþ2mjωj
p;ΛUV−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV−2mjωj
pÞ
du: ðB17Þ
To ha e a non i ial in eg a ion ange o u, i s in eg a-
ion bounda ies should addi ionally sa is y
ΛUV þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV −2mjωj
q≥max P; 2mjωj
P;ðB18Þ
while he o he wo a gumen s o he maximum unc ion in
he lowe in eg a ion bounda y o ua e always smalle han
he uppe in eg a ion bounda y. Equa ion (B18) imposes
cons ain s on he Pin eg a ion. Thus, we change he
bounda ies o he Pin eg a ion acco ding o
Zpþq
jp−qj
dP→Zmin ðpþq;ΛUVþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV−jp2−q2j
pÞ
max ðjp−qj;jp2−q2j=ðΛUVþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Λ2
UV−jp2−q2j
pÞÞ
dP:
ðB19Þ
Fu he mo e, uand Pin eg a ions a e pe o med only i he
lowe in eg a ion bounda ies a e smalle han he uppe
ones. O he wise, hese in eg als a e se o ze o.
R. WALZ, K. BOGUSLAVSKI, and J. BERGES PHYS. REV. D 97, 116011 (2018)
116011-20

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