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Massless posi i i y in g a i on exchange
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Published e sion
He e o-Valea, Ma io; San os-Ga cia, Raquel; Toka e a, Anna
He e o-Valea, M., San os-Ga cia, R., & Toka e a, A. (2021). Massless posi i i y in g a i on
exchange. Physical Re iew D, 104(8), A icle 085022.
h ps://doi.o g/10.1103/phys e d.104.085022
2021
Massless posi i i y in g a i on exchange
Ma io He e o-Valea ,1,2,* Raquel San os-Ga cia,3,†and Anna Toka e a4,5,6,‡
1SISSA, Via Bonomea 265, 34136 T ies e, I aly and INFN Sezione di T ies e, 34127 T ies e, I aly
2IFPU—Ins i u e o Fundamen al Physics o he Uni e se Via Bei u 2, 34014 T ies e, I aly
3Depa amen o de Física Teó ica and Ins i u o de Física Teó ica, IFT-UAM/CSIC, Uni e sidad Au ónoma
de Mad id, Ciudad Uni e si a ia de Can oblanco, 28049 Mad id, Spain
4Depa men o Physics, Uni e si y o Jy äskylä, P.O. Box 35 (YFL), FIN-40014 Jy äskylä, Finland
5Ins i u e o Nuclea Resea ch o Russian Academy o Sciences, 117312 Moscow, Russia
6Helsinki Ins i u e o Physics (HIP), Uni e si y o Helsinki, P.O. Box 64, 00014 Helsinki, Finland
(Recei ed 22 Decembe 2020; accep ed 29 Sep embe 2021; published 27 Oc obe 2021)
We o mula e posi i i y bounds o sca e ing ampli udes including exchange o massless pa icles. We
gene alize he s anda d cons uc ion h ough dispe sion ela ions o include he p esence o a b anch cu
along he eal axis in he complex plane o he Maldes am a iable s. In gene al, alidi y o hese bounds
equi es he cancella ion o di e gences in he o wa d limi o he ampli ude, p opo ional o −1and logð Þ.
We show ha his is possible in he case o g a i ons i one assumes a Regge beha io o he ampli ude a
high ene gies below he Planck scale, as p e iously sugges ed in he li e a u e, and ha he conc e e UV
beha io o he ampli ude is uniquely de e mined by he s uc u e o IR di e gences. We hus ex end
p e ious esul s by including a subleading loga i hmic e m, which we show o be uni e sal. The bounds
ha we p esen he e ha e he po en ial o cons aining e y gene al models o modi ied g a i y and
e ec i e ield heo ies o ma e coupled o g a i a ion.
DOI: 10.1103/PhysRe D.104.085022
I. INTRODUCTION
Posi i i y bounds [1–5] ha e become s anda d ools in
assessing he alidi y o low-ene gy e ec i e ield heo ies
(EFT). By in oking he plausible exis ence o an ul a iole
(UV) comple ion sa is ying easonable p ope ies such as
Lo en z in a iance, uni a i y, and locali y, posi i i y bounds
exclude la ge egions o he pa ame e space o a gi en EFT
by demanding he posi i i y o a ce ain combina ion o
couplings.
In pa icula , hese bounds a e ob ained by combining
he knowledge o he analy ic s uc u e o 2- o-2 sca e ing
ampli udes wi h he op ical heo em
ImAðs; 0Þ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−
4m2
s
σðsÞ;ð1Þ
which ensu es posi i i y o he imagina y pa o he
sca e ing ampli ude Aðs; Þin he o wa d limi →0.
Applica ions o posi i i y bounds include he p oo o
he a- heo em [6,7], he s udy o chi al pe u ba ion heo y
[8], e ec i e Higgs models [9], quan um g a i y [10,11],
massi e g a i y and Galileons [12–16], highe spins [17],
cosmology [18–21], s ing heo y [22,23], and many mo e.
Recen ly, a gene aliza ion o posi i i y bounds, named
a cs, was p oposed [24].
Howe e , all hese examples omi an impo an case o
physical ele ance, he exchange o massless pa icles. In
ha case, he sca e ing ampli ude con ains pa hologies
ha impede one om aking he o wa d limi —a pole −1
and a loga i hmic di e gence logð Þ, due o exchange and
p oduc ion o massless pa icles. This is pa icula ly
ele an in he p esence o g a i y, since g a i ons couple
o all o ms o ma e . Al hough o ene gies below he
Planck scale g a i y could be igno ed, i s cha ac e as a
long ange o ce p oduces con ibu ions o he sca e ing
ampli ude down o he deep in a- ed (IR). Fo mally, he
pa hologies which come wi h he exchange o g a i ons
a e ne e absen and cas a shadow on he alidi y o
posi i i y bounds. E en i one us s he decoupling limi
and he alidi y o g a i y-less posi i i y bounds, i would
be desi able o ind a way o ex end hem o include
g a i on exchange. The e ha e been p e ious a emp s o
sol e his issue by compac i ying space- ime down o
h ee dimensions, whe e g a i ons decompose in massi e
ields [25,26], bu a gene al o malism applicable in mo e
*[email p o ec ed]
†[email p o ec ed]
‡ oka e[email p o ec ed].ac. u
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, 085022 (2021)
2470-0010=2021=104(8)=085022(9) 085022-1 Published by he Ame ican Physical Socie y
a ied si ua ions, wi hou equi ing compac i ica ion, is
s ill lacking.
Recen ly, i was sugges ed ha o wa d di e gences in
g a i on exchange can be cancelled by assuming a Regge
o m o he high-ene gy limi o he sca e ing ampli ude
[27], which is expec ed o hold om s ing heo y [28,29].
Howe e , in [27] only he e m −1is cancelled and no hing
is said abou he loga i hm. This is impo an , hough,
because due o c ossing symme y, equi alen logðsÞand
logðuÞ e ms a e expec ed o coexis in he sca e ing
ampli ude. These e ms spli he complex plane in sin
wo, wi h a b anch cu along he eal line o →0. This
obs uc s he o mula ion o usual posi i i y bounds, which
equi e one o de o m an in eg a ion con ou c ossing he
eal axis.
In his pape we cons uc new posi i i y bounds o
heo ies wi h exchange o massless pa icles, p o ided ha
we cancel he di e gences in he o wa d limi . We show
ha his is indeed possible when he massless s a es
co espond o g a i ons. Gene alizing he esul s o [27],
we p o e ha bo h he pole −1and he logð Þcan be
elimina ed, wi h he emaining pieces in he ampli ude
sa is ying a posi i i y bound eminiscen o he s anda d
case. Finally, we discuss he obus ness o ou esul by
showing ag eemen wi h p e ious wo ks in he li e a u e,
o mally de i ing he bounds ecen ly p oposed by [25,26].
II. DISPERSION RELATIONS
F om now on we will conside ab →ab sca e ing
ampli udes which include a massless pa icle coupled o
hebosonicex e nals a esaand b. The p esence o his
massless s a e will p oduce poles in s, ,andu om ee-
le el exchange, as well as loga i hmic cu s logðsÞ,logð Þ,
and logðuÞindica ing pa icle p oduc ion, ound a loop
le el in pe u ba ion heo y. He e s, ,andua e he
Maldes am a iables, wi h s he ene gy in he cen e -
o -mass ame squa ed. ucan always be elimina ed by
using sþ þu¼4m2, whe e we ha e assumed ha bo h
s a es aand bha e hesamemassm. F om Cauchy’s
in eg al heo em, one can w i e a amily o dispe sion
ela ions o he ampli ude
Aðs; Þ¼ðs−μÞn
2πiIγs
dz Aðz; Þ
ðz−sÞðz−μÞn;ð2Þ
wi h n≥1. The in eg a ion con ou γsmus be aken as a
small ci cle su ounding only he poin z¼s, while he
poin z¼μis a bi a y p o ided ha i lays ou side he
con ou . We ake μ eal he eina e .
A key poin in de i ing posi i i y bounds lays on he
beha io o he sca e ing ampli ude a high ene gies. Fo
massi e pa icles, i can be p o en ha i sa is ies he
F oissa -Ma in bound [30], which implies
lim
jsj→∞
Aðs; Þ
s2
¼0; <4m2:ð3Þ
Alas, he o mal p oo o his bound canno be applied o
he exchange o massless pa icles. Howe e , we will
assume ha his is s ill ue o he cases conside ed he e.
We will jus i y his assump ion la e .
Now we ake he o wa d limi o (2). In he case o
massless pa icles in he in e media e channel, his is
di e gen and canno be aken exac ly. We hus ins ead,
in mo e gene ali y,1expand he ampli ude a ound he limi
→0−
Aðs; 0−Þ≡Aðs; Þj →0−
¼ ðsÞ
þgðsÞlogð ÞþA∘ðsÞþOð Þ;ð4Þ
whe e he limi is aken om he nega i e side o he
eal line.
He e ðsÞand gðsÞa e holomo phic unc ions. When he
sca e ing ampli ude is compu ed in pe u ba ion heo y,
ðsÞcon ains he esidue on he pole o he massless
p opaga o , while gðsÞis p opo ional o he β unc ion o
he ab →ab coupling. The analy ic s uc u e o Aðs; 0−Þis
he e o e con olled by A∘ðsÞ. This is analy ic in he whole
complex plane excep o a b anch cu unning o e he
whole eal line, due o p oduc ion o massless pa icles and
c ossing symme y [30].
The b anch cu obs uc s he s anda d de i a ion o
posi i i y bounds, which uses a con ou in eg al c ossing
he eal line [2]. He e ins ead we no e ha o any eal alue
o s, we can pe o m wo di e en analy ic con inua ions o
he ampli ude, by adding a small imagina y pa siϵ
which mo es he poin o he uppe (down) pa o he
complex plane. A e wa ds we can de o m he in eg a ion
con ou o un abo e (below) he eal axis plus a semi-
ci cum e ence a in ini y, as shown in Fig. 1. This allows
one o de ine wo di e en ealisa ions o (2)
Aðsþiϵ;0−Þ¼ðs−μÞn
2πiZ∞
−∞
dz Aðzþiϵ;0−Þ
ðz−sÞðz−μÞn;ð5Þ
Aðs−iϵ;0−Þ¼ðs−μÞn
2πiZ−∞
∞
dz Aðz−iϵ;0−Þ
ðz−sÞðz−μÞn;ð6Þ
whe e he ci cles a in ini y anish due o (3) and ϵmus be
unde s ood as in ini esimal. He e we keep i ini e only on
nonholomo phic ems. Sub ac ing bo h ep esen a ions we
ge
1This o m encodes all he cases o ele ance o ou knowl-
edge. Fo exchange o scala s and ec o s, bo h ðsÞand gðsÞa e
cons an , while o g a i ons, hey beha e as ∼s2.
HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021)
085022-2
Aðsþiϵ;0−Þ−Aðs−iϵ;0−Þ
¼ðs−μÞn
2πiZ∞
−∞
dz Aðzþiϵ;0−ÞþAðz−iϵ;0−Þ
ðz−sÞðz−μÞn:ð7Þ
In he physical egion s∈Rand we ha e
Aðs−iϵ; Þ¼Aðsþiϵ; Þ. Then
ImAðsiϵ;0−Þ¼∓ðs−μÞn
2πZ∞
−∞
dz ReAðziϵ;0−Þ
ðz−sÞðz−μÞn:
ð8Þ
We now ake Aðsþiϵ;0−Þand use his esul o ew i e
i as
Aðsþiϵ;0−Þ−iImAðsþiϵ;0−Þ
¼ReAðsþiϵ;0−Þ¼ðs−μÞn
2πZ∞
−∞
dz ImAðzþiϵ;0−Þ
ðz−sÞðz−μÞn:
ð9Þ
This exp ession is eminiscen o he s anda d de i a ion
o posi i i y bounds. Howe e , in ou case we ha e
cancelled ou he imagina y pa o he ampli ude, ge ing
id o he discon inui y explici ly.
The in eg al in (9) uns o e nonphysical alues o z.
This can be sol ed by spli ing i in h ee in eg als o e
−∞;0g, 0;4m2gand 4m2;∞g. Pe o ming a change o
a iables z→−zþ4m2in he i s one and using c ossing
symme y, (9) can be ew i en as
Bðs; 0−Þ¼ðs−μÞn
2πZ∞
4m2
dzImAðzþiϵ;0−Þ
ðz−sÞðz−μÞnþð−1ÞnImA×ðzþiϵ;0−Þ
ðz−4m2þsÞðz−4m2þμÞn;ð10Þ
whe e we ha e de ined
Bðs; 0−Þ¼ReAðsþiϵ;0−Þ
−ðs−μÞn
2πZ4m2
0
dz ImAðzþiϵ;0−Þ
ðz−sÞðz−μÞn:ð11Þ
He e A×ðs; 0−Þ¼Að−sþ4m2;0−ÞþOð Þis he c ossed
ampli ude in he uchannel.
Now, by using he op ical heo em (1) in he igh -hand
side o (10), we would be emp ed o ollow he s anda d
de i a ion o posi i i y bounds, and conclude ha
1
n!
dn
dsnBðs;0−Þjs¼0
¼Z∞
4m2
dz
2πzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−
4m2
z
sσðzÞ
znþ1þð−1Þnσ×ðzÞ
ðz−4m2Þnþ1>0;ð12Þ
o e en n, a e aking de i a i es in bo h sides o (10).
None heless, his is no possible in he case a hand.
Ba ing aside he issue o he o wa d limi di e gences—
which we will discuss la e —we mus no e ha he le -
hand side o (10) can be IR di e gen in pe u ba ion heo y.
Fo ini e masses, he in eg al piece in (11) akes ca e o
hese IR di e gences, eplacing hem by m−2and logðm2Þ.
Howe e , his will no wo k in he massless case. To
ci cum en his issue, we ake (10) and de ine ins ead he
ollowing unc ion:
ΣðjÞ¼1
2πiIγδ
ds s3Bðs; 0−Þ
ðs2þδ2Þ2jþ1ð13Þ
whe e now n¼2jand δhas dimensions o ene gy squa ed.
Using now (10) we ind
ΣðjÞ¼Z∞
4m2
dzFðjÞðzÞ;ð14Þ
FIG. 1. In eg a ion con ou s in he complex plane o s.The
zigzag line ep esen s he b anch cu . Fo poin s siϵ,
he in eg a ion con ou γsin he co esponding hal o he
complex plane is shown in ed. The equi alen con ou s used in
(5) a e do ed in blue. The adius o he la ge semici cum e -
ences γ
∞is jsj→∞.
MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021)
085022-3
we e we ha e pe o med he in eg al in sexplici ly, ge ing
FðjÞðzÞ¼z3ImAðzþiϵ;0−Þ
2πðz2þδ2Þ2jþ1
þðz−4m2Þ3ImA×ðzþiϵ;0−Þ
2πððz−4m2Þ2þδ2Þ2jþ1:ð15Þ
No ice ha all dependence on μhas cancelled a e
in eg a ion, wi h δ aking i s place in he denomina o s. The
con ou γδis he sum o wo small ci cles enclosing he
poin s s¼iδ, wi h δ>0. I ac s as a so o an IR
egula o , bu i is no cons ained o be small. No e also ha
he igh -hand side o (14) is posi i e de ini e o all j, since
FjðzÞ>0wi hin he in eg a ion egime om applica ion o
(1). In ha case, we can conclude ha
ΣðjÞ¼1
2πiIγδ
ds s3Bðs; 0−Þ
ðs2þδ2Þ2jþ1>0;ð16Þ
ega dless o he shape o he sca e ing ampli ude, which
migh be e en unknown abo e a ce ain ene gy scale Λ.
Indeed, le us assume ha Aðs; Þis known only wi hin
an EFT wi h alidi y up o2E∼Λ≫m,δ. In ha case,
we can spli he in eg al on he igh -hand side and
ew i e (13) as
ˆ
ΣðjÞ¼Z∞
Λ2
dzFðjÞðzÞ;ð17Þ
whe e
ˆ
ΣðjÞ¼ΣðjÞ−ZΛ2
4m2
dzFðjÞðzÞ:ð18Þ
Again, he igh -hand side o (17) is posi i e and we
conclude
ˆ
ΣðjÞ>0:ð19Þ
Exp essions (16) and (19) a e he massless e sion o
posi i i y and beyond posi i i y bounds [3].Theys a e
ha he con ou in eg al in (13)—o he quan i y ¯
ΣðjÞin
(17)—which can be compu ed in an EFT p o ided ha i
is alid below Λ, has o be posi i e. They di e om
s anda d posi i i y bounds in wo manne s, which encode
he pa icula i ies o he massless exchange. Fi s , we ind
ha he imagina y pa o he ampli ude in (9) cancels ou
om he le -hand side, lea ing only i s eal pa . Second,
he de ini ion o Bðs; 0−Þalso includes an in eg al in he
egion 0≤s≤4m2, which egula es IR di e gences o
massi e ex e nal ields, ensu ing ini eness o he physi-
cal esul . No ice ha when only massi e modes a e
exchanged, ou bounds educe i ially o he s anda d
bounds in he absence o massless poles.
F om now on, all exp essions can be equi alen ly used
wi h ei he ΣðjÞo
ˆ
ΣðjÞ, he only di e ence being he lowe
limi o he in eg al in he igh -hand side o he
dispe sion ela ion. Howe e , i s explici posi i i y does
no change. P o ided ha he ampli ude Aðs; 0−Þis ini e,
hese bounds a e applicable and can lead o in e es ing
cons ain s on he s uc u e o EFT Lag angians h ough
he p esence o he Wilson coe icien s in ReAðs; 0−Þ.
III. REGULARITY IN THE FORWARD LIMIT
Al hough he bounds (16) and (19) a e comple ely
gene al and alid in he case o massless pa icles in he
spec um o he heo y, hey a e meaningless in he
p esence o di e gences in he o wa d limi , as is he case
when g a i ons a e exchanged in he channel.
In ha case, he le -hand side o he bound is domina ed
by he ee-le el con ibu ion o he exchange, which is o
he o m
Aðs; Þ∝−R×s2
;ð20Þ
whe e Ris he esidue in he pole o he g a i on
p opaga o .
Since he limi →0−is con inuous—al hough di e -
gen —in p inciple we can always use (16) o ix he sign o
he di e gence and conclude ha
R>0;ð21Þ
which ells us ha in o de o ag ee wi h uni a i y equi e-
men s, he g a i on mus no be a ghos . Al hough i is
in e es ing o see his i ial condi ion o uni a i y a ising
in his way, he in o ma ion ha i p o ides is sca ce. I we
wan o ex ac mo e in o ma ion om he posi i i y
bounds (16) and (19) in he p esence o a g a i on in
he spec um, hen we need o ind a way o egula ize he
o wa d limi di e gences.
In [27] i is shown3 ha his is possible o he pole −1i
one akes a seemingly s ong assump ion abou he sca e -
ing ampli ude— ha i akes he Regge o m [31]
ImAðs; Þ¼ ð Þðα0sÞ2þlð Þ1þζ
logðα0sÞþO1
α0s;
ð22Þ
which we ex end he e wi h a subleading co ec ion, abo e a
ce ain ene gy scale E∼M. He e ð Þencodes in o ma ion
2No e ha Λmigh no be s ic ly he cu o o he heo y, bu
he ene gy a which he EFT is no a good app oxima ion o he
UV comple e heo y anymo e. This migh happen a ew o de s o
magni ude below he cu o .
3No e howe e ha he bounds de i ed in [27] do no ake in o
accoun he p esence o he b anch cu a all.
HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021)
085022-4

abou he pola iza ion o ex e nal s a es, while lð Þis
cons ained o be nega i e lð Þ<0and sa is ies
lð0Þ¼0, in o de o he ampli ude o uni a ize a high
ene gies. The scale α0is con olled by he alue o he
Regge scale Mas α0∼Oð1ÞM−2
. No e ha in [27]
he loga i hmic co ec ion ha we include he e is no
conside ed.
By assuming his beha io o he ampli ude a la ge s,i
can be easily shown ha he igh -hand side o (17) will also
p esen di e gences when →0, which a e hus con olled
only by he la ge slimi o he in eg al. These can hen be
cancelled agains hose in he le -hand side, wi h he
emaining ini e piece sa is ying i s own e sion o posi i i y.
Re aining only he leading e m in he ampli ude allows one
o cancel he pole −1bu lea es he loga i hmic di e gence
un ouched. As we will see in a momen , he subleading
co ec ion ha we ha e included accoun s o he la e .
O cou se, a his poin one could ques ion he alidi y o
he high-ene gy beha io (22). So a his is an assump ion o
ou wo k, bu one which is well jus i ied in he case o
g a i ons o wo di e en easons. Fi s , le us gi e a
heu is ic a gumen . I we belie e ha s ing heo y p o ides a
UV comple ion o g a i a ional in e ac ions, hen i can be
shown ha g a i on media ed sca e ing ampli udes sa is y
(22) a Oð1Þ,whe eα0is he s ing scale [28,29,32].This
happens due o he con ibu ion o he owe o massi e
modes in he spec um ha a e exci ed abo e hese ene gies.
Loga i hmic co ec ions o simila o m o he ones in (22)
can also be ound in ce ain cases [28] and we expec hem o
a ise om s ing loops.4Second, i ins ead we assume an
a bi a y subleading co ec ion gðsÞ, i can be checked ha
he only choice ha allows o cancelling he loga i hmic
di e gence is p ecisely gðsÞ¼ζ=logðα0sÞ,asi isshownin
Appendix. The e o e, om now on we assume (22).No e
ha by assuming (22), he bound (3) is au oma ically
sa is ied.
We can hen spli he in eg al on he igh -hand side o
(17) in wo
Z∞
Λ2
dzFðjÞðzÞ¼ZM2

Λ2
dzFðjÞðzÞþZ∞
M2

dzFðjÞðzÞ:ð23Þ
Calling Δ¼R∞
M2
dzFðjÞðzÞand using he Regge o m o
he sca e ing ampli ude (22), we ge
Δ¼ ð Þα02þlð Þ
πZ∞
M2

dzz3þlð Þ−4j1þζ
logðα0zÞ;ð24Þ
which can be compu ed explici ly in e ms o he analy ic
con inua ion o he Gamma unc ion.
The o wa d limi can be aken in his exp ession a e
in eg a ion. No e ha since lð0Þ¼0, →0−, and lð Þ<0,
we ha e
lð Þ¼l0ð0Þ þl00ð0Þ
2 2þOð 3Þ;ð25Þ
wi h l0ð0Þ>0. We hus ge
lim
→0−
Δ¼ ð0Þα02
π8
<
:
ððM2
Þ4−4j
4j−4þζα04j−4Γ½0;ð4j−4ÞlogðM2
α0ÞÞ;j>1
ð1
l0ð0Þ −l00ð0Þ
2l0ð0Þ2þlogðM2
α0ÞÞ −ζðγþlogð Þþlog ½−l0ð0ÞlogðM2
α0ÞÞ;j¼1
;ð26Þ
up o e ms which anish when ¼0. He e γis he Eule -
Masche oni cons an , Γðs; xÞ¼R∞
xd s−1e− is he incom-
ple e Gamma unc ion, and we ha e aken z≫m2;δ.We
ha e also assumed ha ou ex e nal s a es sa is y
ImA×ðs; Þ¼ImAðs; Þ om c ossing symme y, which
limi s he applica ion o ou esul o bosonic
s a es. Fe mions will in oduce ex a signs om c ossing
symme y.
We ind ha indeed he leading e m in (22) p oduces a
pole −1, while he subleading co ec ion gi es a logð Þ.
Howe e , no e ha hey only exis when j¼1, while o
j>1 he esul is comple ely egula . This is exac ly he
same kind o di e gence ha we ind in he o wa d limi o
ΣðjÞ, only p esen o j¼1as well.5Thus, we can expand
bo h sides o exp essions (13) and (17) in he limi →0−
and cancel di e gences in he le -hand side agains hose in
he igh -hand side p o ided by Δ, wi h he es o he e ms
emaining ini e. I is pa icula ly in e es ing o no e ha
assuming Regge beha io , which is expec ed o a ise in
g a i y, p ecisely allows o cancella ion o hose di e -
gences p oduced in g a i on sca e ing. As discussed in
Appendix, his seems o be a unique esul .
Explici ly, using (26) we can now ew i e (17) o
j¼1as
4Highe loop con ibu ions like logðlog Þa e expec ed beyond
one loop in he sca e ing ampli ude. We expec hem o cancel
agains highe loop co ec ions in he s ing heo y.
5The di e gen pa o he ampli ude o g a i on exchange is
p opo ional o s2. Thus, i anishes om ΣðjÞwi h j>1a e
e alua ion o he esidues in he pole.
MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021)
085022-5
ˆ
Σð1Þ
R¼ZM2

Λ2
dzFð1ÞðzÞþ ð0Þα02logðM2
α0Þ
π
− ð0Þα02
π
l00ð0Þ
2l0ð0Þ2− ð0Þα02ζ
πlog ½−l0ð0ÞlogðM2
α0Þ
− ð0Þα02ζγ
π;ð27Þ
whe e we ha e in oduced he egula ized e sion o
ˆ
Σð1Þas
ˆ
Σð1Þ
R¼ˆ
Σð1Þþ ð0Þα02
πζlogð Þ−
1
l0ð0Þ ;ð28Þ
by aking all di e gen e ms o he le -hand side.
By choosing he app op ia e alue o he combina ions
ð0Þα02=l0ð0Þand ð0Þα02ζ, he o wa d limi di e gences
can be cancelled, so ha (27) emains egula . No e ha ,
since α02>0and l0ð0Þ>0 his also ixes he sign o ζ
uniquely, al hough in a case by case way.
Finally, we u n ou a en ion o he explici o m o (27).
No e ha he in eg al along Λ2<s<M
2
mus emain
posi i e by applica ion o (1). Howe e , he es o he e ms
do no ha e a de ini e sign. In pa icula , we canno
de e mine he o e all sign o he igh -hand side in (27)
wi hou knowing he alue o l00ð0Þ, which we do no know.
Ne e heless, all hese e ms come mul iplied by he o e all
scale ð0Þα02. Thus, wha we can do is o assess ha he
igh -hand side is posi i e up o he o de in which hey
become impo an . Meaning
ˆ
Σð1Þ
R>−Oð ð0Þα02Þ;ð29Þ
so ha a small amoun o posi i i y iola ion is allowed and
con olled by he dynamics o he UV deg ees o eedom.
Fo j>1 hings a e simple . Since (26) is always
con e gen in his case, he e is no need o expand no
o spli he ange o in eg a ion in (24). Thus, we simply
eco e ou esul (19), which emains alid
ˆ
Σðj>1Þ>0:ð30Þ
Exp essions (29) and (30) a e he inal esul s o ou
wo k. They ep esen posi i i y bounds whose le -hand
sides can be compu ed in an EFT, as long as δ<Λ2, and
whose alue is cons ained by ea u es o he high-ene gy
heo y.
IV. GRAVITATING SCALAR FIELD
Now ha we ha e de i ed use ul posi i i y bounds in he
p esence o exchange o g a i ons, le us es hei alidi y
wi h some well-known heo ies o scala ields coupled o
Eins ein g a i y. The i s case ha we examine is a ee
g a i a ing scala ield, wi h ac ion
S¼Zd4xffiffiffiffiffi
jgj
p−R
2κ2þ1
2∂μϕ∂μϕ;ð31Þ
whe e κ2¼8πG¼M−2
P.
In o de o include he b anch cu in o he sca e ing
ampli ude ϕϕ →ϕϕ we mus a leas compu e he i s loop
co ec ion. Combining i wi h he ee-le el ampli ude we
ge , in he o wa d limi and a e eno maliza ion6
Aðs; 0−Þ¼−κ2s2
−
33κ4s2
24π2ðlogðsÞþlogð−sÞÞ
−
33κ4s2
24π2logð Þ:ð32Þ
He e we ha e used he de Donde gauge and se he
eno maliza ion scale μR¼1in he modi ied minimal
sub ac ion scheme. This choice is ha mless since i s alue
always d ops om he esul .
The in eg al in (11) anishes o massless ex e nal ields.
Thus
Bðs; 0−Þ¼ReAðs; 0−Þ¼−κ2s2
−
33κ4s2
24π2logðs2Þ
−
33κ4s2
48π2logð 2Þ;ð33Þ
and om his we can easily use (13) o compu e
Σð1Þ¼−κ2
−
33κ4
24π23
2þlogð Þþlogðδ2Þ;ð34Þ
Σðj>1Þ¼yðjÞκ2
π2δ4j−4;ð35Þ
whe e yðjÞ>0 o all j. He e we ha e decided no o add
he con ibu ion om RΛ2
0dzFðjÞðzÞ, hus wo king wi h ΣðjÞ
ins ead o
ˆ
ΣðjÞ.
Cancelling he di e gences using (28) de e mines
ð0Þα02∼−l0ð0Þκ2and ð0Þα02ζ∼κ4. Thus he bounds ead
−
33κ4
24π23
2þlogðδ2Þ>−Oð ð0Þα02Þ;ð36Þ
yðjÞκ2
π2δ4j−4>0:ð37Þ
The i s bound is howe e meaningless since he le -hand
side is al eady compa able o he subleading e ms in he
igh -hand side. This o bids us o conclude any hing om
Σð1Þ. On he o he hand, he second bound is au oma ically
6The coe icien in on o he loga i hms is gauge dependen .
Howe e , i s sign is uni e sal o he amily o βgauges [33]
explo ed he e.
HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021)
085022-6
sa is ied o all δ, con i ming a i ial s a emen , ha a ee
g a i a ing scala ield is a bona ide heo y up o MP.
V. SCALAR QED
E en mo e in e es ing is o explo e he case o scala
QED wi h a pho on ϕ, an elec on ψ, and an spec a o ield
χ, as sugges ed in [26]. The ac ion is
S¼Zd4xffiffiffiffiffi
jgj
p−R
2κ2þ1
2∂μϕ∂μϕþ1
2∂μχ∂μχ
þ1
2∂μψ∂μψ−
1
2
Λ2ψ2−λΛϕψ2:ð38Þ
A ene gies below he mass o he elec on Λ≪MP,ψ
can be in eg a ed ou , lea ing a gene ic EFT desc ibing
e ec i e in e ac ions be ween he es o he ields
S¼Zd4xffiffiffiffiffi
jgj
p−R
2κ2þ1
2∂μχ∂μχþ1
2∂μϕ∂μϕ
−λ3Λ
ð2πÞ2
ϕ3
3! þλ4
2π2
ϕ4
4! þDλ2κ2
Λ2ð∂ϕÞ4
þCλ2κ2
Λ2ð∂μϕ∂μχÞ2þ…;ð39Þ
whe e he do s indica e u he Λo κ2supp essed e ms. In
ma ching bo h ac ions, he Wilson coe icien s Dand C
mus be de e mined by a di ec compa ison o a sca e ing
ampli ude. Howe e he e we a e in e es ed in explo ing
wha posi i i y can say abou hem. Following [26] we
ocus on ϕχ →ϕχ, whose one-loop ampli ude gi es
Aðs; 0−Þ¼−κ2s2
þs2Cλ2κ2
Λ2−
11κ4s2
24π2logð Þ
−
11κ4s2
24π2ðlogðsÞþlogð−sÞÞ þ Oð Þ:ð40Þ
Again, we ha e added he one-loop co ec ion, wi h
μR¼1, in o de o make he b anch cu explici . F om he e
we ind
Σð1Þ¼κ2
48 −
48
þ48Cλ2
Λ2−
33κ2
π2
−
22κ2
π2logð Þ−
22κ2
π2logðδ2Þ;ð41Þ
Σðj>1Þ¼yðjÞκ4
π2δ4j−4:ð42Þ
Cancelling he o wa d di e gences we ge again
ð0Þα02∼−l0ð0Þκ2, ð0Þα02ζ∼κ4. F om his he bound
Σðj>1Þ>0is au oma ically sa is ied. I also allows us o
dis ega d he loop co ec ions in Σð1Þ, since hey a e
subleading. We hus ge
Cκ2λ2
Λ2>−Oð ð0Þα02Þ:ð43Þ
This esul ag ees wi h ha o [25,26], whe e i is
p oposed om di e en a gumen s. This also p o es he
conjec u e in hei conclusions o new physics equi ed a a
scale ð ð0Þα02Þ−1=4<M
Pin o de o uni a ize he heo y.
This can be seen om he ac ha a di ec ma ching
be ween he EFT (39) and i s pa ial UV comple ion (38)
demands C<0wi h C∼Oð1Þ, which iola es ou bound.
Thus, (38) needs o be comple ed a in e media e ene gies.
VI. CONCLUSIONS
In his pape we ha e de i ed new posi i i y bounds in
he p esence o exchange o massless pa icles be ween
bosonic s a es. They gene alize and o malize p e ious
esul s in he li e a u e. P o ided ha di e gences in he
o wa d limi can be igno ed, ou bounds can cons ain he
alue o Wilson coe icien s and o he couplings in EFTs
o which he exis ence o a plausible uni a y, Lo en z
in a ian , and local UV comple ion is demanded.
We ha e gone u he and shown ha in he case o
exchange o g a i ons, o wa d di e gences can be can-
celled by assuming a Regge beha io o he sca e ing
ampli ude, which is unique i one assumes analy ici y o he
unc ion lð Þ. Al hough peculia , his o m o he ampli ude
has been p e iously ound in he li e a u e on s ing heo y.
This leads o well-de ined bounds which can now be used
in he p esence o g a i y.
We ha e shown how ou bounds wo k in wo simple
examples. A ee g a i a ing scala ield, whe e hey a e
au oma ically sa is ied, and scala QED wi h a spec a o
ield, o which hey demand new physics below he Planck
scale o uni a ize he heo y, as p e iously sugges ed
by [25,26].
These new bounds open up a window o explo e he
heo y space o phenomenological iable heo ies o
(ma e and) g a i y. We belie e ha ou esul s he e ha e
he po en ial o highly cons ain di e en popula models
cu en ly used o in es iga e p ope ies o black hole
physics and cosmology. I would also be in e es ing o
apply hem o he explo a ion o uni a iza ion mechanisms
o g a i on sca e ing [34–36].
ACKNOWLEDGMENTS
We a e g a e ul o B ando Bellazzini, Ja i Se a, and Ina
Timi yaso o discussions andcommen s. Ou wo k has been
suppo ed by he Eu opeanUnion’s H2020ERC Consolida o
G an “G a i y om As ophysical o Mic oscopic Scales”
G an Ag eemen No. GRAMS-815673 (M. H-V.), by he
Spanish FPU G an No. FPU16/01595 (R. S-G.) and by he
Academy o Finland G an No. 318319 (A. T.). The pa o
wo k o A. T. ela ed o ob aining he bounds om imagina y
poleswassuppo edby heRussianScienceFounda ionG an
MASSLESS POSITIVITY IN GRAVITON EXCHANGE PHYS. REV. D 104, 085022 (2021)
085022-7
No. 19-12-00393. We also wish o acknowledge ne wo king
suppo om COST ac ion CA16104 “GW e se."
APPENDIX: UNIVERSALITY OF THE
SUBLEADING CORRECTION
Le us add ess he e he ques ion on he uniqueness o he
subleading co ec ion o he ampli ude in he Regge limi
(22) equi ed o cancel di e gences in he o wa d limi .
Le us s a by no icing again ha he leading e m, which
cancels he −1con ibu ion o he ampli ude in he IR, was
al eady p oposed in ea lie wo ks [27] and can be ob ained
om a closed s ing ampli ude a e ca e ul manipula ion.
In pa icula , he imagina y pa o he s ing ampli ude is
no a egula unc ion o sand ; i has ins ead an in ini e se
o Regge poles ha equi e egula iza ion. He eina e we
will assume ins ead ha he imagina y pa o he Regge
ampli ude ha we conside is egula in bo h a gumen s
when s→∞, →0.
Going back o FðjÞðz; Þ, de ined in (15), le us examine
he in eg al in (24)
Δj¼Z∞
M2

dzFðjÞðz; Þ:ðA1Þ
No e ha his in eg al can gi e a singula i y a →0only i
i is di e gen when ¼0bu ini e o some small ini e .
In pa icula , o j¼1we ob ain
Δ1¼Z∞
M2

dz ImAðz; Þ
z3:ðA2Þ
Since ImAðs; Þis egula a ¼0 om ou assump ion,
we can Taylo expand i a ound his poin
ImAðs; Þ¼ImAðs;0Þþ∂ ImAðs; Þj ¼0 þOð 2Þ;ðA3Þ
in one o one co espondence o he se ies expansion o he
Regge o m (22),
ImAðs; Þ¼ ð Þðα0sÞ2þlð Þð1þgðsÞÞ ∼
¼ðα0sÞ2ð1þgðsÞÞ½ ð0Þþ ð 0ð0Þ−l0ð0ÞlogðsÞ;
ðA4Þ
whe e we ha e assumed he expansion (25).
A his poin we lea e he o m o he subleading
co ec ion gðsÞcomple ely a bi a y. I we demand ha
he esul o Δ1has he co ec di e gen s uc u e we ha e
Δ1¼Z∞
M2

dz
z ð Þz−lð Þð1þgðzÞÞ¼a
þbð ÞþOð1Þ:ðA5Þ
He e bð Þs ands o he emaining di e gen e ms a →0,
which include he one-loop log e m among o he s.
Changing he in eg a ion a iable o log z¼σand plugging
he small expansion on he in eg and, b ings us o he
condi ion
ð ÞZ∞
log M2

dσe−ðl0ð0Þ þOð 2ÞÞσð1þgðσÞÞ
¼a
þbð ÞþOð1Þ:ðA6Þ
The leading e m in he le -hand side can be compu ed
explici ly and shown o cancel he a −1 e m, while o he
es we ha e
ð0ÞZ∞
log M2

dσe−ðl0ð0Þ ÞσgðσÞ¼bð ÞþOð1Þ:ðA7Þ
A e mul iplying by a s ep unc ion unde he in eg al
sign, his becomes a Laplace ans o m. Al hough i
equi es egula iza ion, i s esul is unique and he e o e
he e exis s a single unc ion gðsÞwhich sa is ies his
iden i y. Since gðsÞ¼ζ=logðα0sÞdoes he wo k, we
conclude ha i is he only op ion.
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