Massless positivity in graviton exchange

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/ Massless posi i i y in g a i on exchange © Au ho s, 2021 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. [1] A. Nicolis, R. Ra azzi, and E. T inche ini, J. High Ene gy Phys. 05 (2010) 095; 11 (2011) 128(E). [2] A. Adams, N. A kani-Hamed, S. Dubo sky, A. Nicolis, and R. Ra azzi, J. High Ene gy Phys. 10 (2006) 014. [3] B. Bellazzini, F. Ri a, J. Se a, and F. Sga la a, Phys. Re . Le . 120, 161101 (2018). [4] C. de Rham, S. Mel ille, A. J. Tolley, and S.-Y. Zhou, J. High Ene gy Phys. 03 (2019) 182. [5] C. de Rham, S. Mel ille, A. J. Tolley, and S.-Y. Zhou, J. High Ene gy Phys. 03 (2018) 011. [6] Z. Koma godski and A. Schwimme , J. High Ene gy Phys. 12 (2011) 099. [7] M. A. Lu y, J. Polchinski, and R. Ra azzi, J. High Ene gy Phys. 01 (2013) 152. [8] A. V. Manoha and V. Ma eu, Phys. Re . D 77, 094019 (2008). [9] I. Low, R. Ra azzi, and A. Vichi, J. High Ene gy Phys. 04 (2010) 126. [10] B. Bellazzini, C. Cheung, and G. N. Remmen, Phys. Re . D 93, 064076 (2016). [11] M. Acce ulli Hube , A. B andhube , S. De Angelis, and G. T a aglini, Phys. Re . D 102, 046014 (2020). [12] C. Cheung and G. N. Remmen, J. High Ene gy Phys. 04 (2016) 002. HERRERO-VALEA, SANTOS-GARCIA, and TOKAREVA PHYS. REV. D 104, 085022 (2021) 085022-8