This is a sel -a chi ed e sion o an o iginal a icle. This e sion
may di e om he o iginal in pagina ion and ypog aphic de ails.
Au ho (s):
Ti le:
Yea :
Ve sion:
Copy igh :
Righ s:
Righ s u l:
Please ci e he o iginal e sion:
CC BY 4.0
h ps://c ea i ecommons.o g/licenses/by/4.0/
Uni a i y o Minkowski nonlocal heo ies made explici
© 2021 he Au ho s
Published e sion
Koshele , Alexey S.; Toka e a, Anna
Koshele , A. S., & Toka e a, A. (2021). Uni a i y o Minkowski nonlocal heo ies made explici .
Physical Re iew D, 104(2), A icle 025016. h ps://doi.o g/10.1103/phys e d.104.025016
2021
Uni a i y o Minkowski nonlocal heo ies made explici
Alexey S. Koshele 1and Anna Toka e a2,3,4
1Depa amen o de Física, Cen o de Ma emá ica e Aplicações (CMA-UBI),
Uni e sidade da Bei a In e io , 6200 Co ilhã, Po ugal
2Depa men o Physics, Uni e si y o Jy äskylä, P.O. Box 35 (YFL), FIN-40014, Jy äskylä, Finland
3Ins i u e o Nuclea Resea ch o Russian Academy o Sciences, 117312 Moscow, Russia
4Helsinki Ins i u e o Physics (HIP), Uni e si y o Helsinki, P.O. Box 64, 00014, Helsinki, Finland
(Recei ed 26 Ma ch 2021; accep ed 22 June 2021; published 23 July 2021)
In his wo k we explici ly show ha he pe u ba i e uni a i y o analy ic in ini e de i a i e scala
ield heo ies can be achie ed using a modi ied p esc ip ion o compu ing sca e ing ampli udes. The c ux
o he new p esc ip ion is he analy ic con inua ion o a esul ob ained in he Euclidean signa u e o he
Minkowski ex e nal momen a. We in ensi ely elabo a e an example o a nonlocal ϕ4model o a ious
in ini e de i a i e ope a o s. Gene al UV p ope ies o ampli udes in nonlocal heo ies a e discussed.
DOI: 10.1103/PhysRe D.104.025016
I. INTRODUCTION
Highe -de i a i e heo ies appea in di e en con ex s
[1–21] om pu e phenomenological models o undamen-
al heo ies wi h he no able example o s ing ield heo y
[22–25] which belongs o he pa icula in e es ing class o
models ea u ing analy ic in ini e de i a i e (AID) ope -
a o s en e ing Lag angians and ac ing as o m ac o s.
Analy ici y o hese ope a o s a low momen a gua an ees a
smoo h local IR limi and p omp s o he UV modi ica ion
o espec i e heo ies.
A model example p o iding such a highe -de i a i e
modi ica ion can be eadily w i en as ollows:
S¼Zd4x−
1
2ϕð□−m2Þ ð□Þ−1ϕ−λ
4! ϕ4ð1Þ
whe e □is he d’Alembe ian ope a o . This ac ion is he
subjec o ou conside a ion in his pape which is aimed a
illus a ing explici ly ha he uni a i y can be main ained in
his class o models. The chosen model ep esen s pe haps
he simples model ea u ing he p ope ies which ough
o be s udied and unde s ood. Namely, he p opaga o is
mani es ly o an in ini e o de in de i a i es and he e is an
in e ac ion e m which gene a es a bounded om below
po en ial. These a e he ea u es ypical o nonlocal models
ha ha e applica ions o quan um g a i y ( eno malizable
and ghos - ee AID heo ies) [19] and UV ini e nonlocal
scala heo ies wi h an a bi a y po en ial [20,26].I is
impo an and in e es ing o pe o m simila compu a ions
in models mo e closely ela ed o he AID g a i y heo ies,
i.e., o o he spins and mo e gene al in e ac ions, bu his is
beyond he scope o he cu en s udy.
Ha ing he me ic signa u e ixed as ðþ−−−Þ, he
abo e Lag angian desc ibes a no mal nonghos ield
p o ided ¼1, which is also no a achyon as long as
m2>0. The p esence o ex a de i a i es may gene ically
spoil he model by he appea ance o ghos s in he
spec um. A simple way o a oid his is o demand ha
ð□Þ−1be an exponen o an en i e unc ion. Then he
sys em has no new deg ees o eedom as he e a e no ini e
poles o he p opaga o apa om he al eady exis ing one
a poin m2. This pic u e seems o be good especially gi en
ha he highe -de i a i e ac o can easily p o ide be e
con e gence o loop in eg als.
He e, exac ly a se ious issue a ises because an exponen
o an en i e unc ion mus ha e an essen ial singula i y a
complex in ini y making he use o he Wick o a ion
unjus i ied [27]. Indeed, he la e equi es ha he p opa-
ga o has a pole a he complex in ini y, no an essen ial
singula i y, and his canno be achie ed in ou model.
The si ua ion, howe e , has a esolu ion p esen ed in he
pape by Pius and Sen [26] o he class o AID heo ies
o igina ing om he s ing ield heo y. The esolu ion
consis s o ce ain modi ica ions o he p esc ip ion o
how he loop in eg als a e compu ed. Namely, specially
designed in eg a ion con ou s o in eg als o e he loop
momen a a e p esc ibed. I is in e es ing o no e ha a
ela ed idea was p esen ed many yea s ago by E imo [2].
To be ph ased in sho , he idea o E imo is o pe o m all
he inne loop in eg als assuming he Euclidean signa u e
o all he momen a. A e a esul is ob ained, one should
Published by he Ame ican Physical Socie y unde he e ms o
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,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 104, 025016 (2021)
2470-0010=2021=104(2)=025016(9) 025016-1 Published by he Ame ican Physical Socie y
con inue his esul analy ically o he compu ed ampli-
ude o Minkowski signa u e o all ex e nal momen a.
This p ocedu e was claimed o gain uni a y sca e ing
ampli udes.
In he p esen wo k, using he simples example o he
ish diag am in ϕ4 heo y, we illus a e ha he idea by
E imo in ac leads o he same de ini ion o he ampli ude
as he me hod elabo a ed by Pius and Sen.1The la e
p o ides ules o how o accoun o he poles a he
complex plane when in eg a ing o e loop momen a. These
ules a e ac ually he same as in local heo ies, since
nonlocal ac o s do no lead o new poles. Le us also
men ion ha he de ini ion o ampli udes by he o mal
applica ion o he amilia Wick o a ion (which is he same
as doing he analy ic con inua ion o he esul upon
comple ion o loop compu a ions) yields a well-de ined
local limi . In his limi , he no mal Wick o a ion is
eco e ed, as well as he uni a i y o a local heo y.
In ou analysis, we i s p o ide an explici example o
compu a ions ixing ð□Þ¼expðα□Þ ha al eady com-
p ises a highly non i ial se o o mulas and hen mo e on
by showing ha one can in eg a e ou in e nal momen a a
one-loop le el o an a bi a y o m ac o ð□Þ. This leads
us o an ex ensi e discussion on UV p ope ies o he
ampli udes in nonlocal heo ies. We hen end up wi h a
conclusion and ou look.
II. FATE OF THE WICK ROTATION
In a local quan um ield heo y, he ampli udes can be
de ined in Minko ski space- ime wi hin he amilia ule
o he poles. Then i is con enien o do a Wick o a ion
and go o Euclidean momen a. In nonlocal heo y, howe e ,
i is o en p oblema ic o de ine he Minkowski ampli ude
because he co esponding in eg als may di e ge. In his
sec ion we show ha he e is no easible way o choose a
special nonlocal o m ac o which would allow o
ob aining con e gen Minkowski ampli udes.
Le us s a wi h he simples adpole co ec ion o he
p opaga o depic ed in Fig. 1.
Fo mally, in Minkowski space we can w i e
A¼Zd4k ðk2
0−

k2Þ
k2
0−

k2−m2:ð2Þ
Then, we can do he Wick o a ion and ob ain he Euclidean
in eg al,
AE¼−iZd4kE ð−k2
0E−

k2Þ
k2
0Eþ

k2þm2ð3Þ
whe e k0E¼−ik0. This in eg al can be made mani es ly
con e gen due o an app op ia e choice o he
unc ion ð□Þ. Howe e , we canno conclude ha A¼AE
unless we check ha he in eg al o e an in ini e a c
AC¼Zπ
2
0
d3kReiθdθ ðR2e2iθ−

k2Þ
R2e2iθ−

k2−m2ð4Þ
anishes in he limi R→∞. In his limi ,
A∞
C¼Zπ
2
0
Reiθdθ ðR2e2iθÞ
R2e2iθ
¼ZR
i ð−z2Þ
z2þZR
ðz2Þ
z2þZπ
2
0
ð0Þe−iθ
dθ
¼ZR
ð0Þðiþ1Þ− ðz2Þ−i ð−z2Þ
z2:ð5Þ
I ð□Þ¼1 hen his in eg al is ze o which makes he Wick
o a ion consis en in a local heo y. Howe e , in nonlocal
models he si ua ion is less ob ious. I A∞
C¼cons <∞
hen AC¼RA∞
Cd3
kis di e gen which means ha he
Minkowski ampli ude is di e gen oo. In p inciple, i is
possible o choose ð□Þin such a way ha he condi ion
Zð ð0Þðiþ1Þ− ðz2Þ−i ð−z2ÞÞdz
z2¼0ð6Þ
is sa is ied. Howe e , when conside ing o he diag ams,
we ob ain o he condi ions, inally an in ini e se o simila
condi ions on ð□Þ, which a p io i would no be esol ed
simul aneously. We hus conclude ha i is nea ly impos-
sible o de ine nonlocal ampli udes in Minkowski space in a
s anda d way unless a mys e ious combina ion o a o m
ac o and he heo y po en ial can be ound, such ha an
in ini e owe o condi ions allowing he Wick o a ion is
esol ed. Independen ly o he o m o he nonlocal
p opaga o , he amilia Wick o a ion canno be done.
The e o e, ano he way o de ine physical nonlocal sca e -
ing ampli udes is equi ed.
III. UNITARITY OF THE FISH DIAGRAM
As we ha e shown jus be o e, he de ini ion o physical
ampli udes in nonlocal heo ies is a icky poin which mos
likely canno be esol ed using s anda d local ield heo y
p esc ip ions by jus making an app op ia e choice o he
nonlocal p opaga o . We hus u n o he me hod elabo a ed
in pape s [2,26].
FIG. 1. Tadpole one-loop con ibu ion o he p opaga o in ϕ4
heo y.
1See also [28,29].
ALEXEY S. KOSHELEV and ANNA TOKAREVA PHYS. REV. D 104, 025016 (2021)
025016-2
We a e going o s udy he p ocedu e using he so-called
ish diag am (See Fig. 2) in nonlocal heo y (1). This
diag am depends on ex e nal momen a in a combina ion
p¼p1þp2bu choosing he cen e o mass e e ence
ame we can educe he numbe o a iables o only one
scala p0. The co esponding Euclidean ma ix elemen
would be de ined as
ME¼−λ2
32π4IðpEÞ:ð7Þ
The physical ma ix elemen is o be de ined as an analy ic
con inua ion o MEðpEÞ o Minkowski momen a p.
He e IðpEÞis de ined as he Euclidean in eg al
IðpEÞ¼Zd4kE
ðk2
EÞ ððpE−kEÞ2Þ
ðk2
Eþm2ÞððpE−kEÞ2þm2Þ:ð8Þ
The in eg al can be ew i en as
IðpEÞ¼2Z ðk2
EÞ ððpE−kEÞ2Þd4kE
p0Eðk2
Eþm2Þðp0E−2k0EÞ:ð9Þ
This in eg al has a pole a k0E¼p0E=2. This esul s in he
nonze o imagina y pa ,
IðpEÞ¼−πiZð ðp2
0E=4þ

k2ÞÞ2d3
k
p0Eð

k2þm2þp2
0E=4Þ
þP:V:2Z ðk2
EÞ ððpE−kEÞ2Þd4kE
p0Eðk2
Eþm2Þðp0E−2k0EÞð10Þ
whe e P:V: ands o he p incipal alue de ini ion o he
in eg al. Le us now con inue analy ically he unc ion
o pE o he physical momen a. Thus, we change
p0E→−ip0,
IðpÞ¼−πZð ð

k2−p2
0=4ÞÞ2d3
k
p0ð

k2þm2−p2
0=4−iϵÞ
−2Z ðk2Þ ðð−ip0−k0Þ2−

k2Þd4k
p0ðk2þm2Þðp0−2ik0Þ:ð11Þ
This in eg al is eal o p2
0<4m2. In he opposi e case, he
i s e m would ha e an imagina y pa , physically
co esponding o he possibili y o a pa icle p oduc ion.
The co esponding exp ession is
ImIðpÞ¼−π3 ðm2Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2
0−4m2
p
p0
θðp2
0−4m2Þ:ð12Þ
Then, aking in o accoun ha nonlocal unc ion should be
no malized in such a way ha ðm2Þ¼1 o p ese e he
alue o he esidue o he p opaga o , we eco e he
amilia esul o he local heo y,
ImM¼λ2
32πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2
0−4m2
pp0
θðp2
0−4m2Þ:ð13Þ
I is hus ob ious ha one obse es he canonical op ical
heo em s a emen which ells us ha he imagina y pa
o some Feynman g aph is equi alen o he p oduc o
exp essions co esponding o pa s in which a gi en g aph
would be cu .
We can demons a e ha he same esul o he ampli-
ude can be ob ained i one akes he o mal de ini ion o
nonlocal ampli ude in Minkowski space,
M¼−iλ2
32π4IðpÞ:ð14Þ
He e
IðpÞ¼Zd4k ðk2Þ ððp−kÞ2Þ
ðk2−m2þiϵÞððp−kÞ2−m2þiϵÞð15Þ
and his in eg al o e Minkowski momen a is di e gen due
o he ine i able p esence o he essen ial singula i y o
unc ion ð□Þand consequen ly some g ow h di ec ion a
in ini y as explained in he p e ious sec ion. Howe e ,
applying o mally he Wick o a ion like in a local heo y
we a i e o he con e gen in eg al. This in eg al can be
w i en in he ollowing o m:
IðpÞ¼−2Z ðk2Þ ððpM−kEÞ2Þd4kE
p0ðk2
E−m2þiϵÞðk0−p0=2þiϵÞð16Þ
whe e we ha e no changed k0 o ik0Ein one ins ance o
accoun o he pole on he eal axis o k0p ope ly. Namely,
we i s ha e o ans o m
1
k0−p0=2þiϵ¼−iπδðk0−p0=2Þ
þP:V:1
k0−p0=2þiϵð17Þ
and hen change k0→ik0Ein he second e m. The o he
pole is no pinched by he con ou . A e ha we a i e a
FIG. 2. One-loop con ibu ion o 2→2sca e ing in ϕ4 heo y.
He e only he s-channel con ibu ion is shown, p¼p1þp2.
UNITARITY OF MINKOWSKI NONLOCAL THEORIES MADE …PHYS. REV. D 104, 025016 (2021)
025016-3
IðpÞ¼πiZd3
kð ðp2
0=4−

k2ÞÞ2
p0ð

k2þm2−p2
0=4−iϵÞ
þ2iZ ðk2
EÞ ððp−kEÞ2Þd4kE
p0ðp0−2ik0EÞðk2
Eþm2Þð18Þ
which di e s om (11) by he ac o −iand his is p ecisely
compensa ed by iin he de ini ion o ampli ude (14).
Thus, he accu a e de ini ion o nonlocal ampli udes
h ough Euclidean in eg als analy ically con inued o
Minkowski ex e nal momen a is equi alen o he amilia
one used o local heo ies. E en hough he la e canno be
o malized h ough he Wick o a ion in nonlocal heo ies
because he Minkowski in eg al is di e gen , his o mal
app oach gi es he same esul s. Fu he , as i was p o en in
[26], he way o he ampli ude’s de ini ion h ough Euclidean
in eg als p ese es he uni a i y o nonlocal heo ies and he
op ical heo em. Also, one can easily see ha he local limi
ð□Þ→1 es o es he s anda d ex book answe s.
IV. FISH DIAGRAM 2-2 SCATTERING
In his sec ion we conside he model wi h A nonlocal
unc ion o he simples o m allowing o he analy ic
compu a ion o he ampli ude, ð□Þ¼eα□. The ma ix
elemen can be o mally de ined as
M¼−iλ2
32π4IðpÞð19Þ
whe e
IðpÞ¼Zd4keαððp=2þkÞ2þðp=2−kÞ2Þ
ððp=2þkÞ2−m2Þððp=2−kÞ2−m2Þ:
ð20Þ
As has been al eady discussed abo e, he Minkowski
in eg al is di e gen and he physical ampli ude is de ined
in ano he way. We eplace loop momen a by hei
Euclidean coun e pa s as we would do o he Wick
o a ion in a local heo y and hen ob ain IðpÞas a sum
o he Euclidean in eg al and he pole pa ,
IðpÞ¼IEðpÞþIpoleðpÞ:ð21Þ
The pole pa he e s ands o he po ion o he whole
exp ession which co esponds o poles which appea a
poin s k0¼p0=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2þm2
pas long as hese poles lie
in he i s and hi d qua e o he in eg a ion con ou plane
upon he p esc ip ion m2→m2−iϵ.
Le us s a wi h he compu a ion o he Euclidean pa ,
IE¼ieαp2=2Zdk0d3ke−2αðk2
0þ

k2Þ
ðk2
0þ

k2þm2−p2=4þipk0Þðk2
0þ

k2þm2−p2=4−ipk0Þ
¼2πieαp2=2Zd dθ 3sin2θe−α 2
ð 2þm2−p2=4Þ2þp2 2cos2θ
¼4π2ieαp2=2
p2Z d e−2α 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð 2þm2−p2=4Þ2þp2 2
ð 2þm2−p2=4Þ
s−1!:ð22Þ
In he massless limi ,
IE¼4π2ieαp2=2
p2Zp2=4−ϵ0
0
dze−2αzz
p2=4−z
þZ∞
p2=4þϵ0
dze−2αzp2=4
z−p2=4:ð23Þ
E alua ing hese in eg als one ob ains he con ibu ion o
ReM
IRe
E¼−
2π2i
p2αðeαp2=2−1−2αp2Eiðαp2=2Þ
þαp2ðγþlog ϵ0ÞÞ:ð24Þ
He e γis Eule cons an , EiðzÞis he in eg al exponen , and
we ha e aken he mean alue o he in eg als pe o ming
he limi ϵ0→0). As we will see la e , he di e gence log ϵ0
will be exac ly canceled by he pole pa o he ampli ude in
he massless limi .
The pole pa is nonze o only when he e a e poles in I
and III sec o s o he complex plane, i.e., when
k2þm2>p
2=4. Figu e 3illus a es his si ua ion. In his
case, hei impac o he ampli ude is gi en by he sum o
he ele an esidues in (20)
Ipole ¼−2πiZd3kResk0¼p0=2−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2þ

k2
q
−Resk0¼−p0=2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2þ

k2
q ð25Þ
which eadily simpli ies o
ALEXEY S. KOSHELEV and ANNA TOKAREVA PHYS. REV. D 104, 025016 (2021)
025016-4

Ipole ¼−2πiZd3ke2αm2þ2αpðp=2−ffiffiffiffiffiffiffiffiffiffi

k2þm2
pÞ
2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2þ

k2
pðp=2−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2þ

k2
pÞ
:ð26Þ
Fo m¼0we ob ain
Ipole ¼−
8π2ieαp2
p2Zp2−ϵ0
0
zdze−4αz
p2=4−z:ð27Þ
We no ice ha in eg a ion up o p2 ollows om he
ac ha k2þm2should be g ea e han p2in o de o
ha e poles in he I and III qua e s o he in eg a ion plane.
Ipole ¼2π2i
αp2ðeαp2
−1−αp2Eiðαp2Þþαp2ðγþlog ϵ0ÞÞ
ð28Þ
Summing up and adding he compu ed in Sec. III imagina y
pa con ibu ion −π3 o IðpÞwe ge he inal answe 2
IðpÞ¼−π3þ2iπ2
αp2eαp2
−αp2Eiðαp2Þ−eαp2=2
þ1
2αp2Eiðαp2=2Þ:ð29Þ
The eal pa o ampli ude (19) is plo ed in Fig. 4whe e eh
p esence o a s ong coupling egime seems ob ious.
We will see in he nex sec ions ha his si ua ion depends
on he o m ac o .
One can u he obse e, ha o small a gumen s
EiðzÞ≈γþlogðzÞþz, he local limi and he loga i hmic
singula i y usually a ising in he cu o egula iza ion
es o e as long as α→0.
The o al sca e ing ampli ude including he c ossed
diag ams can be ob ained as
M o ¼−iλ2
32π4ðIðffiffiffis
pÞþIðffiffi
pÞþIðffiffiffi
u
pÞÞ ð30Þ
whe e s anda d Mandels am a iables a e used.
V. COMPUTATION FOR A GENERIC
PROPAGATOR
The compu a ion o he p e ious sec ion can be ex ended
o a gene ic p opaga o , i.e., gene ic o m ac o ð□Þ. This
can be done by u ilizing he x ep esen a ion o p opa-
ga o s. This yields, in ou case, he ollowing qui e simple
exp ession o he ma ix elemen :
MðpÞ¼ λ2
2ð2πÞ4Zd4xΔðxÞ2eipx þiλ2π
32
þλ2
16pZp=2
−p=2
ð2pqÞdq ð31Þ
whe e ΔðxÞis he p opaga o in he x ep esen a ion and he
whole o mula is, in a sense, he Fou ie ans o m o he
p oduc o he p opaga o s. In he i s e m bo h Fou ie
ans o m and in eg a ion o e xa e pe o med in he
Euclidean space. The las e m co esponds o he pole
con ibu ion and i has such a simple o m only o e en
o m ac o s ð−k2Þ¼ ðk2Þ. They a e o a special in e es
because hey would simul aneously decay along bo h eal
and imagina y di ec ions in k.
The p opaga o in xspace can be easily w i en as
ΔðxÞ¼ i
16π4Zd4k ðk2Þ
k2eikx ð32Þ
FIG. 3. Poles o he ampli ude (20).
-4 -2 2 4
- 0.0002
0.0002
0.0004
FIG. 4. Ampli ude ( eal pa ) dependence on he ex e nal
momen um. He e α¼1,m¼0,λ¼0.1.
2The eal pa o his ampli ude was also compu ed in [30].
UNITARITY OF MINKOWSKI NONLOCAL THEORIES MADE …PHYS. REV. D 104, 025016 (2021)
025016-5
and in con as wi h a local heo y, he la e in eg al can be
s aigh o wa dly made well de ined a sho dis ances, i.e.,
small x. Fi s , we go o sphe ical coo dina es and simpli y
he las exp ession as ollows:
ΔðxÞ¼ i
4πZ∞
0
dk ðk2ÞJ1ðkxÞ
xð33Þ
whe e J1ðzÞis he Bessel unc ion.
Then we ge by di ec subs i u ion
MðpÞ¼−λ2
64π3pZ∞
0
J1ðpxÞJ1ðkxÞJ1ðqxÞ ðk2Þ ðq2Þdkdqdx þiλ2π
32 þλ2
32p2Zp2
−p2
ðzÞdz: ð34Þ
One can pe o m an in eg a ion o e xin he i s in eg al in (34) analy ically and he esul is as ollows [31,32]:
Z∞
0
J1ðpxÞJ1ðkxÞJ1ðqxÞdx ¼(1
2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðp2−ðk−qÞ2ÞððkþqÞ2−p2Þ
p
pkq o jk−qj<p<kþq
0o he wise:
One u he is le wi h an in eg a ion o e kand qin he
i s in eg al in (34). The domain o in eg a ion can be
a bi mo e in ui i ely ew i en as −kþp<q<kþp;
0<k<pg∪ k−p<q<kþp; k > pg.
The las e m in (34) demons a es a no e y much
expec ed p ope y. Fo la ge p, as long as he o m ac o is
an in eg able unc ion, his e m alls uni e sally as ∼1=p2,
i espec i e o a pa icula o m ac o . We will see below
ha o a la ge class o sui able unc ions, his makes
he leading and appa en ly uni e sal con ibu ion o he
ampli ude.
P oceeding by compu ing he ampli udes nume ically
o di e en o m ac o s ðk2Þwe choose he ollowing
wo examples o a nonlocal unc ion. As he i s example
we ake3
1ðk2Þ¼e−k4ð35Þ
being he simples nonlocal unc ion allowing o s aying in
he pe u ba i e egime [33]. Besides ha , we conside a
mo e in ol ed unc ion which alls polynomially along he
eal axis while i is an exponen o an en i e unc ion. This
unc ion was i s sugges ed by Tomboulis in [8]
2ðk2Þ¼e−Γð0;k4Þ−γ−log k4:ð36Þ
He e Γð0;k
4Þis an incomple e gamma unc ion. The la e
o m ac o , ha ing a polynomial decay a in ini y, makes
he heo y a oid a possible s ong coupling egime. Fo
la ge eal k, one has 2ðk2Þ∝1=k4. In Fig. 5we plo he
absolu e alue o his unc ion on he whole complex plane
o make i s g ow h di ec ions isible. Also, we plo he
Euclidean ou -dimensional Fou ie images Δ1ðxÞand
Δ2ðxÞ o he unc ions 1ðkÞand 2ðkÞ, espec i ely,
in Fig. 6.
The esul ing ampli ude ( eal pa ) nume ically com-
pu ed wi h he use o (34) is plo ed in Fig. 7. One can see
ha he cu es all o momen a much highe han he
nonlocali y scale.4Howe e , as was men ioned abo e, he
asymp o es o his alling a e o he same o de ∼1=p2 o
bo h unc ions and a e de e mined by he las e m in
exp ession (34). This e m a ises due o he pole s uc u e in
Minkowskian ampli udes. Fu he , one can deduce nume i-
cally ha he beha io o he i s e m in (34) depends
on he UV beha io o he nonlocal p opaga o and di e s
o he wo cases unde conside a ion. Fo unc ion 1i is
an exponen ial all as ∼e−p4while o unc ion 2i is he
powe -law supp ession ∼p−6. We also see ha he model
emains weakly coupled o bo h chosen o m ac o s in
con as wi h he eαp2 o m ac o discussed in he p e ious
sec ion.
VI. UV PROPERTIES OF THE AMPLITUDES
IN NONLOCAL THEORY
The UV beha io o he sca e ing ampli udes is de e -
mined by he o m o he nonlocal p opaga o . As we
ob ained in Sec. IV, o he p opaga o e□=Λ2=□ he
ampli ude is g owing exponen ially o posi i e p2. Does
i mean ha his kind o model is no alid om he poin
o iew o quan um ield heo y [34]? We canno immedi-
a ely conclude ha such ypes o p opaga o s canno be
conside ed because a ound he nonlocali y scale he model
unde goes a s ong coupling egime which means ha one-
loop app oxima ion is no alid anymo e. Howe e , his
3He ea e in ou nume ical s udies we se he nonlocal scale o
be uni y.
4The cu es also g ow o small momen a, ep oducing he
loga i hmic di e gence in he local limi bu his g owing is no
appa en o he chosen domain o he nume ic in eg a ion.
ALEXEY S. KOSHELEV and ANNA TOKAREVA PHYS. REV. D 104, 025016 (2021)
025016-6
does no mean ha he uni a i y is b oken as long as
nonpe u ba i e me hods can, in p inciple, lead o a esul
which is s ill consis en wi h uni a i y. We lea e his
ques ion o u u e s udy.
In his wo k we concen a e on he esul s which can
be ob ained pe u ba i ely. I he p opaga o ðk2Þ=k2is
alling o bo h signs o k2 hen we expec he loop
con ibu ions o all o all physical alues o ex e nal
momen a. Howe e , he sca e ing ampli ude would ine i-
ably g ow o some di ec ions o sand in he complex
plane. Le us check he consis ency o such a beha io wi h
he known esul s abou he p ope ies o he ampli udes.
Fo example, he Ma in-F oissa bound [35] ells us ha
he ampli ude in he o wa d limi →0is bounded by
jMðs; 0Þj <Csðlog sÞ2ð37Þ
on he scomplex plane. Al hough his bound was ob ained
in local heo ies, he ecen wo k [34] p o ides a o mal
p oo o he simila bound in nonlocal heo ies,
jMðs; 0Þj≲Cs1þ2a:ð38Þ
This bound was ob ained wi hin a he weak assump ions
including he alidi y o pa ial wa e expansion, uni a i y,
analy ici y, and he exponen ial boundedness jMðs; Þj <
sNeajsja o ixed . He e α,N,Ca e some posi i e
cons an s. These assump ions, excep he las one, a e
expec ed o be ul illed in he models we conside in his
wo k and in mo e gene ic se ings like in [26]. The bound
(38) ac ually means ha he ampli udes in he o wa d
limi a e polynomially bounded while one would expec he
exponen ial g ow h in some di ec ions on he complex
splane.
Le us discuss his issue using he simple model wi h a
cubic in e ac ion whe e he exponen ial beha io shows up
al eady a ee le el.
S¼Zd4x1
2ϕð□−m2Þ ð□Þ−1ϕ−λ
3! ϕ3:ð39Þ
The ee le el ampli ude in his model has he o m
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1.0
-10 -5 510
0.05
0.10
0.15
0.20
0.25
0.30
FIG. 6. The le plo ep esen s he unc ions 1ðkÞ;
2ðkÞand he igh plo shows hei Fou ie images Δ1;2ðxÞ.
FIG. 5. Absolu e alue o he unc ion 2ðRe k; Im kÞ.
1 2 3 4
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
FIG. 7. Sca e ing 2-2 one-loop ampli ude as a unc ion o he
ex e nal momen um. The ed cu e is o o m- ac o 1ðkÞand
blue cu e is o o m- ac o 2ðkÞ. He e we ha e aken λ¼0.1.
UNITARITY OF MINKOWSKI NONLOCAL THEORIES MADE …PHYS. REV. D 104, 025016 (2021)
025016-7
Mðs; Þ∼λ2 ðsÞ
s−m2þ ð Þ
−m2þ ðuÞ
u−m2:ð40Þ
Thus, ðsÞmus ha e a di ec ion o exponen ial g ow h
since an en i e unc ion, which is polynomially bounded on
he whole complex plane, is no hing else bu a polynomial
o ini e o de . Howe e , o all physical momen a he
ampli ude is small i we equi e ðq2Þ o all o eal q2.
Thus, wi h his choice, he model s ays in he pe u ba i e
egime o physical alues o momen a. Howe e , i one
compu es he ampli ude o complex alues o momen a
his could lead o b eaking o he pe u ba ion heo y. The
nai e compu a ion leading o he exponen ial g ow h can be
in alid. The e o e, he conclusion abou he ampli ude’s
exponen ial g ow h in he nonphysical egion is no co ec .
The bound (38) leads o u he in e es ing consequences
o he model (39). Unde he condi ion o polynomial
boundedness o he ampli ude in he o wa d limi , we
ha e he Ce ulus-Ma in lowe bound on he ampli ude
alling [36],
jMðs; →0Þj >Ce
−γffiffis
plogðs=s0Þ:ð41Þ
He e C; γ;s
0a e posi i e cons an s. This bound is ob i-
ously iola ed i we ake ðk2Þ∝e−αk4. The ee le el
ampli ude is alling oo as while he model is in a
pe u ba i e egime so we can us his compu a ion o
physical momen a. This con adic ion means ha some-
hing is w ong wi h he model wi h an exponen ially alling
p opaga o . No ice ha in he wo k [8] he choice o
polynomially supp essed p opaga o s was sugges ed o
be mo i a ed by di e en a gumen s such as powe coun -
ing, o example. The p opaga o alling as 1=knis s ill
consis en wi h he discussed bounds.
No ice ha dis a o ing he exponen ially supp essed
p opaga o is no applicable o models wi h con ac ϕ4
in e ac ions since he leading e m is gi en by a cons an
ee le el con ibu ion. This kind o supp ession o hose
models is no in a con adic ion wi h Ma in-Ce ulus
bound [36].
VII. CONCLUSION AND OUTLOOK
We ha e shown explici ly ha he op ical heo em
s a emen holds in nonlocal highe -de i a i e heo ies upon
u ilizing a modi ied compu a ion p esc ip ion. The p e-
sc ip ion consis s o compu ing Feynman g aphs, assuming
all he in e nal momen a a e Euclidean and hen con inuing
he inal answe analy ically o Minkowski signa u e o
ex e nal momen a. Resul ing exp essions sa is y he op ical
heo em s a emen and obey he co ec local heo y limi .
We we e able o show ha he one-loop co ec ion o he
e ex in a nonlocal ϕ4 heo y can be compu ed o
a bi a y o m ac o s and ha e p esen ed bo h analy ic
and nume ic de i a ions o he esul . We ound ha ,
independen o he nonlocal o m ac o , he leading e m
in he ampli ude a high ene gies alls as ∼1=p2as long
as ðk2Þ¼ ð−k2Þ. This e m a ises as a esul o he
compu a ion o he pole con ibu ion o he ampli ude.
Thus, nonlocal ampli udes o e en o m ac o s a one loop
ha e a uni e sal beha io in he UV limi .
As an impo an obse a ion we ha e ound ha unde an
assump ion ha he Ma in-Ce ulus bound [36] is as
sa is ied in nonlocal models as in local ones, hen he
ampli udes should no all oo as obeying bound (41).I
such a nonlocal heo y has a con ac in e ac ion hen his
bound can always be sa is ied i espec i ely o he decay
a e o he p opaga o as he con ac in e ac ion e m always
domina es in he pe u ba i e expansion. On he o he hand,
in he absence o he con ac in e ac ion ( o ins ance
compu ing co ec ions o 2-2 sca e ing in ϕ3 heo y)
one mus demand ha p opaga o decays polynomially
a o ing as such he class o unc ions sugges ed in [8].
As he nex s ep, one would aim a demons a ing ha he
backg ound ield me hod p o ides consis en esul s and
hus can be applied o AID models which is a plausible
expec a ion. Also he s ong coupling egime, which may
a ise in such nonlocal ield heo ies [20] due o he g ow h
o he ampli udes o complex pa ame e s, is an impo an
ques ion o u he in es iga ions. Ye ano he ambi ious
p oblem is o s udy such heo ies in he la ge-Napp ox-
ima ions [37] and see how hey a e compa ed o hei local
coun e pa s.
ACKNOWLEDGMENTS
The au ho s would like o hank I. A e ’e a and M.
Shaposhniko o commen s and ques ions on he manu-
sc ip . A. K. is suppo ed by FCT Po ugal in es iga o
P ojec No. IF/01607/2015. A. T. is suppo ed by he
Academy o Finland G an No. 318319. The pa o he
wo k pe o med by A. T. ela ed o he nume ic compu a-
ions o he ampli udes is suppo ed by he Russian Science
Founda ion G an No. 19-12-00393.
No e added.—Upon comple ion o he manusc ip we ha e
me wi h g ea in e es a ela ed s udy [38] discussing he
same p oblem o uni a i y o nonlocal ield heo ies.
ALEXEY S. KOSHELEV and ANNA TOKAREVA PHYS. REV. D 104, 025016 (2021)
025016-8