Paolo Di Vecchia

Soft Theorems for Massless Particles from Gauge Invariance

Paolo Di Vecchia

Niels Bohr Institute, Copenhagen and Nordita, Stockholm

Brussels, 17.05.2018

Infrared Physics: Asymptotics & BMS symmetry, soft theorems, memory, information paradox and all that Brussels 16-18 May 2018

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 1 / 71 This talk is based on the work done together with

I Zvi Bern, Scott Davies and Josh Nohle “Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance", Phys. Rev. D 90 (2014) 084035, arXiv:1406:6987. I Raffaele Marotta, Matin Mojaza and Josh Nohle “The story of the two dilatons", Phys. Rev. D93 (2016) n.8,085015, arXiv:1512.03316.

I Raffaele Marotta and Matin Mojaza “Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string”, JHEP 1512 (2015) 150, arXiv:1502.05258. “Subsubleading soft theorem for gravitons and dilatons in the bosonic string”, JHEP 1606 (2016) 054, arXiv:1604.03355. “Soft behavior of a closed massless state in superstring and universality in the soft behavior of the dilaton”, JHEP 1612 (2016) 020, arXiv:1610.03481. “The B-field soft theorem and its unification with the graviton and dilaton”, JHEP 1710 (2017) 017, arXiv:1706.02961. Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 2 / 71 Plan of the talk 1 Introduction

2 Scattering of a photon and n scalar particles

3 Scattering of a graviton/dilaton and n scalar particles

4 Soft limit of gluon amplitudes

5 Soft limit of (n + 1)-graviton/dilaton amplitude

6 Soft behavior of the Kalb-Ramond Bµν 7 Soft theorems in string theory

8 Origin of string corrections

9 Soft behavior of graviton and dilaton at loop level

10 String dilaton versus ﬁeld theory dilaton

11 Soft theorem from Ward Identities

12 Double-soft behavior

13 Conclusions and Outlook

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 3 / 71 Introduction

I In particle physics we deal with many kinds of symmetries.

I They all have the property of leaving the action invariant, but have very different physical consequences.

I GLOBAL INTERNAL UNBROKEN SYMMETRIES

I Unique vacuum annihilated by the symmetry gener.: Qa 0 = 0 | i I Particles are classiﬁed according to multiplets of this symmetry and all particles of a multiplet have the same mass.

I If mu = md , QCD would be invariant under an SU(2)V ﬂavor symmetry.

I and the proton and the neutron would have the same mass (neglecting the electromagnetic interactions). 2m2 −m2 +m2 −m2 mu π0 π+ k+ k0 I Since = 2 2 2 = 0.56 = 1, SU(2) is only an md m −m +m V k0 k+ π+ 6 approximate symmetry.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 4 / 71 I GLOBAL INTERNAL SPONT. BROKEN SYMMETRIES. 0 I Degenerate vacua: Qa 0 = 0 . | i | i I Not realized in the spectrum, but presence in the spectrum of massless particles, called Nambu-Goldstone bosons. I For zero quark mass, QCD with two ﬂavors is invariant under SU(2) SU(2) . L × R I It is broken to SU(2) = 3 broken generators. V ⇒ I The NG bosons are the three pions in QCD with 2 ﬂavors. I This is one physical consequence of the spontaneous breaking. I Another one is the existence of low-energy theorems. I The scattering amplitude for a soft pion is zero at low energy: (Adler zeroes). I If one introduces a mass term, breaking explicitly chiral symmetry and giving a small mass to the pion, one gets the two Weinberg scattering lengths (ππ ππ): → 7m m = π ; = π a0 2 a2 2 32πFπ −16πFπ

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 5 / 71 I SPACE-TIME GLOBAL SYMMETRIES

I Conformal invariance is the most notable example.

I It is a classical symmetry if the action does not contain any dimensional quantity = Symmetry of the tree diagrams. ⇒ I In general, it is broken by anomalies in the quantum theory.

I Introduction of the dimensional quantity µ in the ren. process.

I It can be also explicitly broken by, for instance, mass terms.

I It can be spontaneously broken by, for instance, non-zero vacuum expectation value of a scalar ﬁeld.

I As a consequence, one gets a NG boson, called the dilaton that has a universal low-energy behavior: only one NG boson.

I In general, in the full quantum theory, the dilaton gets a mass proportional to the β-function of the theory.

I In = 4 super Yang-Mills, it stays massless in the quantum th.. N I In this talk we derive the universal soft dilaton behavior from the conformal WT identities.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 6 / 71 I LOCAL INTERNAL AND SPACE-TIME SYMMETRIES I A local internal symmetry requires the introduction of massless gauge bosons with spin 1. I A local space-time symmetry requires the introduction of a massless spin 2 graviton. I In both cases, local gauge invariance is necessary to reconcile the theory of relativity with quantum mechanics. I It allows a fully relativistic description, eliminating, at the same time, the presence of negative norm states in the spectrum of physical states.

I Although described by Aµ and Gµν, both gluons and gravitons have only two physical degrees of freedom in D=4. I Another consequence of gauge invariance for photons is charge conservation, while for gravitons is momentum conservation. I Yet another physical consequence of local gauge invariance is the existence of low-energy theorems for photons and gravitons [F. Low, 1958; S. Weinberg, 1964]

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 7 / 71 I The interest on the soft theorems was revived few years ago by [Cachazo and Strominger, arXiv:1404.1491[hep-th]].

I They study the behavior of the n-graviton amplitude when the momentum q of one graviton becomes soft (q 0). −1 ∼ I The leading term O(q ) was shown to be universal by Weinberg in the sixties, 0 I They suggest a universal formula for the subleading term O(q ).

I They speculate that, as the leading term, it may be a consequence of BMS symmetry of asymptotically ﬂat space-times.

I This has been claimed later, in four space-time dimensions, to be a consequence of the BMS Ward-Takahashi identities.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 8 / 71 I In this talk we show that the conditions:

µν µν qµMn+1(q; ki ) = qνMn+1(q; ki ) = 0 completely determine the terms of (q−1), (q0) and (q1) of µν O O O the symmetric part of Mn+1(q; ki ) in terms of the amplitude without the soft particle. −1 0 I They also determine the terms of( (q ), (q )) of the µν O O (antisymmetric) part of Mn+1(q; ki ) in terms of the amplitude without the soft particle. I q is the momentum and µ, ν are indices of the soft particle.

I The soft particle can be a graviton, a dilaton or a Bµν field, while the other particles are massive scalars or massless gravitons, dilatons and Bµν fields. I The amplitude for a soft graviton, dilaton and Bµν field is then µν obtained by saturating Mn+1 with their respective polarizations. I This procedure can be easily extended to massive particles with any spin.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 9 / 71 I The above conditions fix the soft behavior of the Kalb-Ramond 0 −1 field Bµν only up to the order q (no term of order q and terms of order q are not fixed by gauge invariance).

I The soft behavior obtained in this way is valid not only for the tree diagrams but also for loop diagrams when the theory containing these soft particles is free from UV and IR divergences.

I May be also when there are IR divergences. See below.

I The soft behavior obtained in this way is conﬁrmed by explicit calculations in string theory (with possibly the addition of string corrections).

I The string corrections are naturally explained by the structure of the three-point amplitudes.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 10 / 71 I It turns out that the soft behavior of the gravity dilaton includes the generators of scale and special conformal transformations.

I Then, we consider a conformal theory in D dimensions where the generators of scale and special conformal transformations are spontaneously broken.

I A massless Nambu-Goldstone boson appears that in the literature is also called dilaton.

I This dilaton has, in principle, nothing to do with the previous dilaton.

I We then show that the Ward-Takahashi identities of scale and special conformal transformations imply a low-energy behavior (at the tree level) that is very similar to that of the gravity dilaton.

I We ﬁnally discuss similarities and differences.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 11 / 71 One photon and n scalar particles

(a) (b)

µ I The scattering amplitude Mn+1(q; k1 ... kn), involving one photon and n scalar particles, consists of two pieces:

n µ µ X ki Mn+1(q; k1,..., kn) = ei Mn(k1,..., ki + q,..., kn) ki q i=1 · µ + Nn+1(q; k1,..., kn) .

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 12 / 71 I and must be gauge invariant for any value of q: n µ X qµMn+1 = ei Mn(k1,..., ki + q,..., kn) i=1 µ +qµNn+1(q; k1,..., kn) = 0

I Expanding around q = 0, we have n X ∂ 0 = ei Mn(k1,..., ki ,..., kn) + qµ Mn(k1,..., ki ,..., kn) ∂kiµ i=1 µ 2 + qµN (q = 0; k ,..., kn) + (q ) . n+1 1 O I At leading order, this equation reduces to n X ei = 0 , i=1 which is simply a statement of charge conservation [Weinberg, 1964]

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 13 / 71 I At the next order, we have n µ X ∂ qµNn+1(0; k1,..., kn) = ei qµ Mn(k1,..., kn) . − ∂kiµ i=1 µ I This equation tells us that Nn+1(0; k1,..., kn) is entirely determined in terms of Mn up to potential pieces that are separately gauge invariant. I It is easy to see that the only expressions local in q that vanish µ under the gauge-invariance condition qµE = 0 are of the form,

µ µν µν νµ E = qνA ; A = A − where Aµν is an antisymmetric tensor (local in q) constructed with the momenta of the scalar particles. I The explicit factor of the soft momentum q in each term means that they are suppressed in the soft limit and do not contribute to µ Nn+1(0; k1,..., kn).

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 14 / 71 I We can therefore remove the qµ leaving

n µ X ∂ Nn+1(0; k1,..., kn) = ei Mn(k1,..., kn) , − ∂kiµ i=1

µ thereby determining Nn+1(0; k1,..., kn) as a function of the amplitude without the photon.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 15 / 71 I Inserting this into the original expression yields

n µ X ei µ µν Mn+1(q; k1,..., kn) = ki iqνLi Mn(k1,..., kn) + (q) , ki q − O i=1 · where µν µ ∂ ν ∂ Li i ki ki ≡ ∂kiν − ∂kiµ is the orbital angular-momentum operator. I The amplitude with a soft photon with momentum q is entirely 0 determined, up to (q ), in terms of the amplitude Mn(k1,..., kn), involving n scalar particlesO and no photon. I This goes under the name of F. Low’s low-energy theorem. I Low’s theorem for photons is unchanged at loop level. I Even at loop level, all diagrams containing a pole in the soft momentum are of the form shown, with loops appearing only in the blob and not correcting the external vertex.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 16 / 71 µ I Get an amplitude by contracting Mn+1(q; k1,..., kn) with the photon polarization εqµ. I Soft-photon limit:

h (0) (1)i M + (q; k ,..., kn) S + S Mn(k ,..., kn) + (q) , n 1 1 → 1 O where

n (0) X ki εq S ei · , ≡ ki q i=1 · n µν (1) X εqµqνLi S i ei , ≡ − ki q i=1 · µν where Li is the orbital angular momentum.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 17 / 71 I Actually from the gauge invariance condition: n X µ ei Mn(k1 ... ki + q ... kn) + qµNn+1(q; k1 ... kn) = 0 i=1 one can get a more general soft theorem. [Z.Z. Li, H.H. Lin and S.Q. Zhang, arXiv:1802.03148 [hep-th]] I Deﬁning ∞ µ X µ;µ1...µ` Nn+1(q; k1 ... kn) = qµ1 ... qµ` N `=0 the gauge inv. condition ﬁxes only the symmetric part of Nµ;µ1...µ` . I One gets n e ∂ ∂ ∂ µ;µ1...µ` X i N (k1 ... kn) = ... − (` + 1)! ∂kiµ ∂kiµ ∂kiµ i=1 1 ` µ;µ1...µ` Mn(k ... kn) + A (k ... kn) × 1 1 µ;µ ...µ` where A 1 (k1 ... kn) is antisymmetric under the exchange of µ with µi .

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 18 / 71 I This ﬁxes Mn+1 to be

n µ P∞ 1 µν ∂ ` ! k i qνJ (q ) µ X i − `=0 (`+1)! i ∂ki Mn+1(q; k1 ... kn) = ei ki q i=1 µ Mn(k ... kn) + A (q; k ... kn) × 1 1 where ∞ µ X µ;µ1...µ` µ A (q; k1 ... kn) = qµ1 ... qµ` A ; A (q = 0; k1 ... kn) = 0 `=1

µ;µ ...µ` I A 1 is antisymmetric exchanging µ with any µi .

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 19 / 71 I An inﬁnite order soft theorem is then obtained:

∂ ∂ µ Ωµµ1...µ` ... Mn+1(q; k1 ... kn) q=0 = ∂qµ1 ∂qµ` | " n ` ∂ ∂ X ei ( i) µν ∂ = Ωµµ1...µ` ... − qνJi q ∂qµ ∂qµ (qki )(` + 1)! ∂ki 1 ` i=1 # Mn(k ... kn) × 1 q=0

where Ωµµ1...µ` is a symmetric matrix. I This is in agreement with the result of [Y. Hamada and G. Shiu, arXiv:1801.05528] obtained with other methods.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 20 / 71 One graviton/dilaton and n scalar particles

I In the case of a graviton scattering on n scalar particles, one can write n µ ν µν X ki ki Mn+1(q; k1,..., kn) = Mn(k1,..., ki + q,..., kn) ki q i=1 · µν + Nn+1(q; k1,..., kn) , µν I Nn+1(q; k1,..., kn) is symmetric under the exchange of µ and ν. I On-shell gauge invariance implies µν 0 = qµMn+1(q; k1,..., kn) n X ν µν = ki Mn(k1,..., ki + q,..., kn) + qµNn+1(q; k1,..., kn) . i=1 I and also µν 0 = qνMn+1(q; k1,..., kn) n X µ µν = ki Mn(k1,..., ki + q,..., kn) + qνNn+1(q; k1,..., kn) . Paolo Di Vecchiai (NBI+NO)=1 Soft behavior Brussels, 17.05.2018 21 / 71 I Before proceeding further let us be more precise.

I In general, gauge invariance implies the more general conditions:

µν µν qµ M (q; k ... kn) f (q; k ... kn)η = 0 n+1 1 − 1 µν µν qν M (q; k ... kn) f (q; k ... kn)η = 0 n+1 1 − 1 where f is an arbitrary function of the momenta.

I Such extra function is irrelevant for a soft graviton because its µν polarization is traceless: q ηµν = 0. I But, it would be relevant for the dilaton!

I It turns out, however, that explicit string calculations (at the tree level) show that this term is present only if the amplitude also includes open strings.

I We will see that this term has an important physical interpretation.

I Preliminary explicit string calculation (at the multi-loop level) in the bosonic string also show that this term is present and has a very simple interpretation.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 22 / 71 I At leading order in q, we then have n X µ ki = 0 , i=1

I It is satisﬁed due to momentum conservation. I Different couplings to different particles would have prevented the leading term to vanish: Gravitons have universal coupling [Weinberg, 1964]. I At ﬁrst order in q, one gets n X ν ∂ µν ki Mn(k1,..., kn) + Nn+1(0; k1,..., kn) = 0 , ∂kiµ i=1

I while at second order in q, it gives

n 2 " µν ρν # X ν ∂ ∂Nn+1 ∂Nn+1 ki Mn(k1,..., kn) + + (0; k1,..., kn) = 0 . ∂kiµ∂kiρ ∂qρ ∂qµ i=1

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 23 / 71 I Using the previous conditions + the corresponding ones (µ ν) (the other particles are scalar particles) one gets ↔

n µ ν µ νρ ν µρ µν X ki ki ki qρLi ki qρLi Mn+1(q; ki ) = κD i i ki q − 2ki q − 2ki q i=1 · · · µρ νσ µ ν σ 1 qρL qσL 1 k q q ∂ i i + ηµνqσ qµηνσ i − 2 k q 2 − − k q ∂k σ i · i · i 2 Mn(k ,..., kn) + O(q ) , × 1 2 (D) where κD = 8πGN . I The previous expression is gauge invariant by construction:

µν µν qµMn+1 = qνMn+1 = 0

I This is precisely the soft-dilaton(graviton) behavior of an amplitude 2 4 with n tachyons (at the tree level) in the bosonic string (mi = α0 ) 0 − with no α corrections ! Bµν is not coupled to only tachyons.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 24 / 71 I We see that the physical-state conditions for the graviton µ ν µν q µν = q µν = 0 ; η µν = 0 set to zero the terms that are proportional to ηµν, qµ and qν. I We are then left with the following expression for the graviton soft limit of a single-graviton, n-scalar amplitude:

h (0) (1) (2)i 2 M + (q; k ,..., kn) S + S + S Mn(k ,..., kn) + (q ) , n 1 1 → 1 O I where n µ ν X εµνk k S(0) i i , ≡ ki q i=1 · n µ νρ X εµνk qρL S(1) i i i , ≡ − ki q i=1 · n µρ νσ 1 X εµνqρL qσL S(2) i i . ≡ −2 ki q i=1 · I These soft factors follow entirely from gauge invariance.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 25 / 71 I On the other hand, if we saturate it with the dilaton polarization:

d 1 2 2 = (ηµν qµq¯ν qνq¯µ); q = q¯ = 0 , qq¯ = 1 µν √D 2 − − − we get

( n 2 d µν κD X mi ρ ∂ µνMn+1(q; k1 ... kn = 1 + q ρ √D 2 − ki q ∂k − i=1 · i n 1 ∂2 X ∂ + qρqσ + 2 k µ 2 ∂k ρ∂k σ − i ∂k µ i i i=1 i ρ 2 2 q µ ∂ ∂ 2 + 2 ki µ ρ kiρ µ Mn + (q ) , 2 ∂ki ∂ki − ∂ki ∂kiµ O

th where mi is the mass of the i scalar particle. I The soft factors follow entirely from gauge invariance.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 26 / 71 Soft limit of n-gluon amplitude

1 n 1

n n 1 n 1 − − (a) (b) (c)

I We consider a tree-level color-ordered amplitude where gluon (n + 1) becomes soft with q k + . ≡ n 1 I Being the amplitude color-ordered, we have to consider only two poles.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 27 / 71 I By introducing the spin contribution to the angular-momentum operator,

µσ (Σ )µi ρ i (ηµµi ηρσ ηµρηµi σ) , i ≡ − we can write the total amplitude as

µ;µ1···µn Mn+1 (q; k1,..., kn) δµ1 k µ iq (Σµσ)µ1 ρ 1 σ 1 ρ ρµ2···µn = − Mn (k1 + q, k2,..., kn) √2(k q) 1 · δµn−1 k µ iq (Σµσ )µn ρ n σ n−1 ρ µ1···µn−2ρ − Mn (k1,..., kn−2, kn + q) − √2(k − q) n 1 · µ;µ1···µn + Nn+1 (q; k1,..., kn) .

I Notice that the spin terms independently vanish when contracted with qµ.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 28 / 71 I On-shell gauge invariance requires

µ;µ1···µn 0 = qµMn+1 (q; k1,..., kn)

1 µ1µ2···µn = Mn (k1 + q, k2,..., kn) √2

1 µ1···µn−1µn Mn (k1,..., kn−1, kn + q) − √2 µ;µ1···µn + qµNn+1 (q; k1,..., kn) .

I For q = 0, this is automatically satisﬁed.

I At the next order in q, we obtain 1 ∂ ∂ µ1···µn Mn (k1, k2 ... kn) −√2 ∂k1µ − ∂knµ µ;µ1···µn = Nn+1 (0; k1,..., kn) .

µ;µ1···µn I Thus, Nn+1 (0; k1,..., kn) is determined in terms of the amplitude without the soft gluon.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 29 / 71 I With this, the total expression becomes

µ;µ1···µn Mn+1 (q; k1 ... kn) k µ µ 1 kn µ1···µn = Mn (k1,..., kn) √2(k q) − √2(kn q) 1 · · q (Jµσ)µ1 σ 1 ρ ρµ2···µn i Mn (k1,..., kn−1) − √2(k1 q) µσ· µn qσ(Jn ) ρ µ1···µn−2ρ + i Mn (k1,..., kn) + (q) , √2(k − q) O n 1 · where

µσ µσ µσ (J )µi ρ L ηµi ρ + (Σ )µi ρ, i ≡ i i with ∂ ∂ µσ µ σ µσ µi ρ µµi ρσ µρ µi σ Li i ki ki ; (Σi ) i (η η η η ) ≡ ∂kiσ − ∂kiµ ≡ −

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 30 / 71 I In order to write the ﬁnal result in terms of full amplitudes, we contract with external polarization vectors.

I We must pass polarization vectors ε1µ1 and εnµn through the spin-one angular-momentum operator such that they will contract ρµ2···µn with the ρ index of, respectively, Mn (k1,..., kn) and µ1···µn−2ρ Mn (k1,..., kn). I It is convenient to write the spin angular-momentum operator as

∂ ∂ µσ µi ρ µ σ ρ εiµi (Σi ) ρM = i εi εi εiρM . ∂εiσ − ∂εiµ

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 31 / 71 I We may therefore write

h (0) (1)i M + (q; k ,..., kn) S + S Mn(k ,..., kn) + (q) , n 1 1 → n n 1 O where

(0) k1 ε kn ε Sn · · , ≡ √2 (k1 q) − √2 (kn q) · µσ · µσ (1) J1 Jn Sn iεµqσ . ≡ − √2 (k q) − √2 (kn q) 1 · ·

I Here

Jµσ Lµσ + Sµσ , i ≡ i i where µν µ ∂ ν ∂ µσ µ ∂ σ ∂ Li i ki ki , Si i εi εi . ≡ ∂kiν − ∂kiµ ≡ ∂εiσ − ∂εiµ

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 32 / 71 Soft limit of (n + 1)-graviton/dilaton amplitude

I As before the amplitude is the sum of two pieces:

µν;µ1ν1···µnνn Mn+1 (q; k1,..., kn) n 1 h i X µ µi α µρ µi α ν νi β µσ νi β = ki η iqρ(Σi ) ki η iqσ(Σi ) ki q − − i=1 · µ1ν1··· ···µnνn M (k ,..., k + q,..., kn) × n αβ 1 i µν;µ1ν1···µnνn + Nn+1 (q; k1,..., kn) , where

µρ (Σ )µi α i (ηµµi ηαρ ηµαηµi ρ) . i ≡ −

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 33 / 71 I On-shell gauge invariance implies

µν;µ1ν1···µnνn 0 = qµMn+1 (q; k1,..., kn) n h i X ν νi β νρ νi β µ1ν1···µi ···µn−1νn−1 = k η iqρ(Σ ) M (k ,..., k + q,..., kn) i − i n β 1 i i=1 µν;µ1ν1···µnνn + qµNn+1 (q; k1,..., kn) . I Proceeding as before we end up getting n µ ν µ νρ ν µρ µν X ki ki ki qρJi ki qρJi Mn+1(q; ki ) = κD i i ki q − 2ki q − 2ki q i=1 · · · µρ νσ µ ν σ 1 qρJi qσJi 1 µν σ µ νσ ki q q ∂ + η q q η σ − 2 ki q 2 − − ki q ∂ki · · 1 qρqσηµν qσqνηρµ qρqµησν ρ ∂ − − i −2 ki q ∂iσ · 2 Mn(k ,..., kn) + (q ) , × 1 O where Mn(k1,..., kn) is the n-point scattering amplitude involving gravitons and/or dilatons Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 34 / 71 I In order to write our expression in terms of amplitudes, we saturate with the soft graviton polarization tensor µν µ ν µν q µν = q µν = η µν = 0

I As for gluons, passing the polarization vectors through the spin operators, we get

h (0) (1) (2)i 2 M + (q; k ,..., kn) = S + S + S Mn(k ,..., kn) + (q ) , n 1 1 n n n 1 O where n µ ν (0) X εµνki ki Sn , ≡ ki q i=1 · n µ νρ (1) X εµνki qρJi Sn i , ≡ − ki q i=1 · n µρ νσ (2) 1 X εµνqρJi qσJi Sn . ≡ −2 ki q i=1 · Same result as from string theory apart from α0 correct. of O(q ).

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 35 / 71 I These soft factors follow entirely from gauge invariance and agree with those computed by Cachazo and Strominger for D = 4. I Remember that Jµσ Lµσ + Sµσ , i ≡ i i with µσ µ ∂ σ ∂ µσ µ ∂ σ ∂ Li i ki ki , Si i εi εi . ≡ ∂kiσ − ∂kiµ ≡ ∂εiσ − ∂εiµ

I Projecting along the dilaton, we get ( n d µν κD X µ ∂ M (q; k1 ... kn) = 2 k µ µν n+1 √D 2 − i ∂k − i=1 i ρ 2 2 q µ ∂ ∂ (i) ∂ 2 ki µ ρ + kiρ µ + 2iSµρ − 2 − ∂ki ∂ki ∂ki ∂kiµ ∂kiµ n ρ σ ) X q q (i) µν (i) ∂ 2 + (Sρµ)η (Sνσ) + Diρ σ Mn + (q ) 2ki q ∂ O i=1 · i Same result as from string theory (no α0 corrections!).

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 36 / 71 Soft behavior of the Kalb-Ramond Bµν

I We start again with the pole term:

n ρ σ ¯ pole X [kiµ iq Sµρ][kiν iq Sνσ] M(n+1)µν = κD − − Mn(ki + q) , ki q i=1 · where ∂ ∂ ¯ ∂ ∂ Sµρ = i iµ ρ iρ µ ; Sνσ = i ¯iν σ ¯iν µ ∂i − ∂i ∂¯i − ∂¯i

I For the antisymmetric part one gets

n σ ¯ ρ ρ σ ¯ pole X iki[µq Si ν]σ iki[νq Si µ]ρ q Si [µρq Si ν]σ M(n+1)µν = κD − − − ki q i=1 · Mn(k + q) , × i 1 where A[µBν] = 2 (AµBν AνBµ). −1 − I No leading term (q ) appears (Weinberg, 1965)

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 37 / 71 I The previous expression is not gauge invariant

I We can add to it a term that will make it gauge invariant obtaining:

n " σ ¯ ρ ρ σ ¯ X iki[µq Si ν]σ iki[νq Si µ]ρ q Si [µρq Si ν]σ Mµν = κD − − − ki q i=1 · i ¯ + S µν S µν Mn(k + q) + Nµν(q; k ) . 2 i − i i i

I In this case gauge invariance imposes:

µ ν q Nµν(q; ki ) = q Nµν(q; ki ) = 0 .

I They are satisﬁed if we impose ∂ N (q = 0; k ) = 0 ; N (q; k ) = A µν i ∂qρ µν i ρµν

Aρµν is a completely antisymmetric tensor.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 38 / 71 I In conclusion, one gets n " σ ¯ ρ ρ σ ¯ X iki[µq Si ν]σ iki[νq Si µ]ρ q Si [µρq Si ν]σ M(n+1)µν = κD − − − ki q i=1 · i ¯ ρ + S µν S µν Mn(k + q) + q Aρµν(q, k ) . 2 i − i i i [R. Marotta, M. Mojaza and PDV, JHEP 1710 (2017) 017, arxiv:1708.02961 [hep-th]] I This is an exact relation between an n + 1 and an n-point amplitude, valid in any order in the soft expansion. I Obviously Aρµν, being gauge inv., cannot be ﬁxed by gauge inv.. I Subsubleading term is not ﬁxed by gauge invariance. I In string theory the subsubleading term cannot be written in a factorized form and contains a term with the Bloch-Wigner Dilog that is gauge invariant by itself. ∞ 1 z X zn 2iD (z) = Li (z) Li (z¯) + log z log − ; Li = 2 2 − 2 − | | 1 z¯ 2 n2 − n=1

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 39 / 71 Soft theorems in string theory

I In string theory the soft theorems have been investigated by Ademollo et al (1975) and J. Shapiro (1975) B.U.W. Schwab, arXiv:1406.4172 and arXiv:1411.6661 M. Bianchi, Song He, Yu-tin Huang and Congkao Wen, arXiv:1406.5155 M. Bianchi and A. Guerrieri, arXiv:1505.05854 and arXiv:1512.00803 + our papers already cited

I One can compute the amplitude with a massless closed string state and other particles and derive the soft behavior.

I The leading and the two sub-leading terms are exactly those derived by imposing gauge invariance apart from string corrections in the subsubleading term.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 40 / 71 I If the soft particle is a dilaton and the other particles are tachyons (at the tree level) one gets:

( n 2 d µν κD X mi ρ ∂ µνMn+1(q; k1 ... kn) = 1 + q ρ √D 2 − ki q ∂k − i=1 · i n 1 ∂2 X ∂ + qρqσ + 2 k µ 2 ∂k ρ∂k σ − i ∂k µ i i i=1 i ρ 2 2 q µ ∂ ∂ 2 + 2 ki µ ρ kiρ µ Mn + (q ) , 2 ∂ki ∂ki − ∂ki ∂kiµ O

where m2 = 4 and i − α0

− Z Qn 2 0 8π κD n 2 d zi Y α i=1 2 ki kj Mn = 0 zi zj α 2π dVabc | − | i6=j

is the correctly normalized n-tachyon amplitude.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 41 / 71 I When the other particles are massless, the soft behavior of a graviton or a dilaton (at the tree level) is given by: n X 1 i ρ M(n+1)µν(q; ki ) = κD kiµkiν q kiµ(Ji )νρ + kiν(Ji )µρ ki q − 2 i=1 µρ νσ µ ν 1 qρJ qσJ k q ∂ i i + i qσ + qµηνσ ηµνqσ −2 k q k q − ∂k σ i i · i 1 qρqσηµν qσqνηρµ qρqµησν ρ ∂ − − i −2 ki q ∂iσ · 0 kiµkiν +α qσkiνηρµ + qρkiµησν ηρµησν(ki q) qρqσ − · − ki q · ρ ∂ 2 i Mn(k1 ... kn) + (q ) × ∂iσ O obtained through explicit calculations on the full bosonic string amplitude and containing α0 correction to the order q.

I String corrections in the heterotic string, but not in superstring.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 42 / 71 I Projecting along the dilaton we get ( n µν κD X µ ∂ Mµν(q; ki ) = 2 k µ d √D 2 − i ∂k − i=1 i ρ 2 2 q µ ∂ ∂ ρ (i) ∂ + 2 ki µ ρ kiρ µ iq Sµρ 2 ∂ki ∂ki − ∂ki ∂kiµ − ∂kiµ n ρ σ ) X q q (i) µν (i) ∂ 2 + (Sρµ)η (Sνσ) + Diρ σ Mn + (q ) 2ki q ∂ O i=1 · i I No string corrections (in the soft operator) for the soft dilaton. I Generators of space-time scale and spec. conf. transf. µ 2 λ ˆ = xµ ˆ ; ˆµ = (2xµxλ x ηµλ) ˆ . D P K − P where ˆµ is the generator of space-time translations. P I Going to momentum space they become: 2 2 ˆ ∂ ˆ ν ∂ ∂ = ikµ ; µ = 2k ν µ kµ ν , D − ∂kµ K − ∂k ∂k − ∂k ∂kν What is the reason for their presence? See below. Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 43 / 71 0 I If we restrict us to (the regular) terms of (q ), for both the bosonic string and superstring, we get: O " n # κD X ∂ Mn+1 q0 = 2 kiµ Mn . | √D 2 − ∂kiµ − i=1 This is the old result of Ademollo et al (1975). I It can be written in a more suggestive way by observing that, in general, Mn has the following form: − 8π κD n 2 1 gs D−3 D−2 √ 0 2 √ 0 2 Mn = 0 Fn( α ki ) , κD = D−10 (2π) ( α ) , α 2π 2 4 √2 where Fn is dimensionless and Mn trivially satisﬁes the condition: n ! 0 ∂ X ∂ D 2 √α + kiµ 2 + (n 2) − Mn = 0 − ∂√α0 ∂kiµ − − 2 i=1 I This equation and the soft behavior imply: κD 0 ∂ D 2 ∂ Mn+1 = √α + − gs Mn + (q) . √D 2 − ∂√α0 2 ∂gs O − Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 44 / 71 I Same result if we include massless open strings (on a Dp-brane), due to the presence of the term proportional to ηµν in the gauge invariance condition.

I The amplitude of a soft dilaton is obtained from the amplitude without a dilaton by a simultaneous rescaling of the Regge slope 0 α and the string coupling constant gs. I Same rescaling that leaves Newton’s constant invariant: 0 ∂ D 2 ∂ √α + − gs κD = 0 − ∂√α0 2 ∂gs

I No fundamental dimensionless constant in string theory.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 45 / 71 I Apply to the case n = 5 with 5 dilatons:

n ! κD X ∂ M5 = 2 kiµ M4 + (q) √D 2 − ∂kiµ O − i=1 where

0 0 0 tu su st Γ(1 α s )Γ(1 α u )Γ(1 α t ) M = κ2 + + − 4 − 4 − 4 4 D s t u α0s α0u α0t Γ(1 + 4 )Γ(1 + 4 )Γ(1 + 4 )

0 I In the ﬁeld theory limit (α 0) (supergravity), one gets zero because one has a homogenous→ function of degree 2.

I This is consistent with the fact that M5 is vanishing in ﬁeld theory, due to the Z2 symmetry. I In string theory one gets a non-trivial right-hand-side.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 46 / 71 I What about the field theory limit of the (q) term? O I The Z2 symmetry of the dilaton in field theory makes all amplitudes with an odd number of dilatons and any number of gravitons vanish (if no B-fields are involved) I Thus, more generally for an amplitude, Mn involving any number of gravitons and an even number of dilatons n ! X ∂ lim 2 kiµ Mn = 0 α0→0 − ∂kiµ i=1

I As shown in [Loebbert, Mojaza, Plefka, arXiv:1802.05999], the soft theorem also implies invariance under special conformal transformation 2 2 ! µ ∂ ρ ∂ µρ ∂ lim k 2k ρ 2iS ρ Mn = 0 α0→0 i 2 i i ∂ki − ∂ki kiµ − ∂ki

I Full conformal invariance can be established by introducing a d−2 multiplicity dependent scaling dimension ∆ = n .

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 47 / 71 Origin of string corrections

I In the closed bosonic string the polarization stripped three-point on-shell amplitude for massless states reads: 0 µν; µ ν ; αβ µ α µ M i i = 2κ ηµµi qα ηµαqµi + ηµi αk k qµi qµ 3 D − i − 2 i α0 ηννi qβ ηνβqνi + ηνi βk ν k νqνi qν × − i − 2 i

I It has string corrections with respect to the ﬁeld theory three-point amplitude. 0 3 0 2 I They come from the Gauss-Bonnet (α ) and R ((α ) ) terms [Zwiebach (1985), Matseev+Tseytlin (1987)]. I If, in the pole terms, one keeps also the string corrections in the three-point amplitude, then, from gauge invariance, one obtains precisely the string corrections in the soft behavior. I String corrections also in the heterotic string but not in superstring. I Universal behavior of a soft dilaton: never string corrections.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 48 / 71 Soft behavior of graviton and dilaton at loop level

With R. Marotta and M. Mojaza, to appear

I The h-loop amplitude involving one graviton/dilaton and n tachyons is equal

n Z Z 0 0 (h) Y α Y α µ n+1 2 2 k`qGh(z,z`) 2 ki kj Gh(zi ,zj ) ν Mn+1 = CgN0 dM d z e e q ¯q `=1 i

[Frau, Lerda, Sciuto, DV, 1987] [Petersen and Sidenius, 1987] [Mandelstam, 1985 and 1992]

I is the h-loop Green function. Gh

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 49 / 71 I In the Schottky parametrization the measure is given by

Qn 2 h 2 2 2 = d zi Y d kad ξad ηa 4 − D = i 1 ( τ) 2 dM 4 4 1 ka det Im dVabc ka ξa ηa | − | a=1 | | | − | 0 " ∞ ∞ # Y Y 1 Y 2D 1 k n 4 |1 k n | | − α| α n=1 − α n=2 In the Schottky parametrization of a Riemann surface the moduli are ka (multiplier) and ξa and ηa (two ﬁxed points) of the Schottky generators Sa. I The same amplitude at the tree level is equal to n Z Qn 2 Z 0 0 d zi Y α Y α n+1 i=1 2 2 k`qG0(z,z`) 2 ki kj G0(zi ,zj ) M0 = C0N0 d z e e dVabc `=1 i

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 50 / 71 I Except for an extra term and the measure, the two amplitudes have the same form in terms of the Green functions. I The two Green functions are of course different. I Using their general properties (2) ∂z¯∂z h(z, w) = πδ (z w) + T (z) Z G − Z 2 2 d z∂z¯∂z h(z) = 0 = d zT (z, w) = π Σh G ⇒ Σh − 0 α k qG(z ,z ) e 2 i i i = 0 we can show that, except for an extra term, the soft theorem is exactly as at the tree level. I A delicate point is that, in the bosonic string, the integral over the moduli is IR divergent. (h) (h) I Our result is valid if we regularise both Mn+1 and Mn with an IR cutoff. I The soft operator plus the extra term will be acting on the h-loop n-tachyon amplitude.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 51 / 71 0 I The extra term is modifying only the (regular) soft term of order q as follows:

n ! (h) κD X ∂ (h) Mn+1 q0 = kiµ + 2 + h(D 2) Mn | √D 2 − ∂kiµ − − i=1

I The h-loop n-tachyon amplitude is equal to

1−h ( − ) (h) 8π κD 2 h 1 1 κD n Mn = 0 hD α 2π (2πα0) 2 2π n Z Z 0 Y 2 Y α k k G(z ,z ) dM d z e 2 i j i j × i i=1 i

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 52 / 71 I It satisﬁes the following condition:

n 0 ∂ X ∂ √α + kiµ h(D 2) 2 − ∂√α0 ∂kiµ − − − i=1 D 2 (h) + − (n + 2(h 1)) M = 0 2 − n

that is equal to

n ! 0 ∂ X ∂ D 2 ∂ √α + kiµ h(D 2) 2 + − gs − ∂√α0 ∂kiµ − − − 2 ∂gs i=1 M(h) = 0 × n

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 53 / 71 0 I Since the subleading (regular) term of (q ) of the soft dilaton amplitude is equal to O

n ! (h) κD X ∂ (h) Mn+1 q0 = kiµ + 2 + h(D 2) Mn | √D 2 − ∂kiµ − − i=1 then, for an arbitrary loop, we have

κ ∂ D 2 ∂ (h) D √ 0 (h) Mn+1 q0 = α + − gs Mn , | √D 2 − ∂√α0 2 ∂gs − precisely as at the tree level! −1 0 1 I The other terms of order q , q , q are as at tree level.

I This has been shown rigorously at one-loop level and preliminary calculations also indicate that it is true at multiloop level.

I Except possibly for the term (q), coming from the extra term, that has been shown to be zeroO only at one-loop.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 54 / 71 String dilaton versus ﬁeld theory dilaton

I We have seen that the low energy behavior of the string dilaton involves the generators of the conformal group ˆ and ˆµ. D K I String theory is not conformal invariant because it contains a dimensional constant α0.

I Why then the soft behavior of the string dilaton contains the generators of scale and special conformal transformations?

I In the literature the word dilaton is used for both the string dilaton and the Nambu-Goldstone boson of spontaneously broken conformal invariance.

I To distinguish them, we call the second one ﬁeld theory dilaton.

I Recently the identiﬁcation of the two dilatons has been proposed in a paper by [R. Boels and W. Wormsbecher, 1507.08162]

I They proposed that the two dilatons have the same soft behavior.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 55 / 71 I This is, in general, only correct at the tree level because, for the ﬁeld theory dilaton, the loop corrections break explicitly conformal invariance.

I There are theories, as = 4 super Yang-Mills, that remain conformal invariant alsoN at the quantum level.

I In these theories we expect the dilaton soft theorems to be valid also at loop level.

I For the string dilaton, instead, the soft behavior is mantained also at the loop level provided that the theory is free from UV and IR divergences (or with an IR cutoff).

I In the following we derive the soft behavior of the ﬁeld theory dilaton that follows from the conformal WT identities.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 56 / 71 I As an example consider a conformal invariant version of the Higgs potential:

2 2 1 µ 1 µ λ 2 2 L = ∂µH∂ H ∂µΞ∂ Ξ H Ξ −2 − 2 − 2 −

I Flat direction corresponding to H = Ξ = a. h i h i I If a = 0 then conformal invariance is spontaneously broken. 6 I In terms of the ﬁelds: Ξ + H Ξ H = r + √2a ; − s √2 √2 ≡

I one gets the following Lagrangian (m = 2√2aλ):

1 µ 1 µ 1 2 2 2 2 2 2 2 L = ∂µr∂ r ∂µs∂ s m s 4√2aλ rs 2λ r s −2 − 2 − 2 − − s is massive and r is a massless dilaton. I What is the soft behavior of a dilaton?

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 57 / 71 I By explicit calculation one gets:  n 2 µ ∂  mi 1 + q µ 1 X ∂ki X ∂ Tn+1  + 4 n ki  Tn + (q) ∼ √ − ki q − − ∂ki O 2a i=1 i

I This result is more general: it follows from the WT identity of the 2 µ dilatational current and from the eq. √2 a ∂ r = Tµ . − I For the string dilaton we get:  n 2 µ ∂  mi 1 + q µ X ∂ki X ∂ Mn+1 κd  + 2 ki  Mn + (q) ∼ − ki q − ∂ki O i=1 i

I The two soft behaviors are similar but not equal. Why? I In the case of a NG boson all dimensional factors are rescaled by D−2 a scale transformation and one gets D n 2 4 n (for D = 4) that is the dimension of the amplitude. − → − 1 I In string theory one rescales only the factor α0 keeping κD ﬁxed.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 58 / 71 Soft theorem from Ward Identities

I We consider a field theory whose action is invariant under some global transformation. µ I We call j the corresponding Nöther current and we consider the following matrix element: ∗ µ T 0 j (x)φ(x ) . . . φ(xn) 0 h | 1 | i I Taking the derivative with respect to x and then performing a Fourier transform, we get: Z D −iq·x h ∗ µ d x e ∂µ T 0 j (x)φ(x ) . . . φ(xn) 0 − h | 1 | i ∗ µ i +T 0 ∂µj (x)φ(x ) . . . φ(xn) 0 h | 1 | i n X −iq·xi ∗ = e T 0 φ(x ) . . . δφ(x ) . . . φ(xn) 0 , − h | 1 i | i i=1 where δφ is the infinitesimal transformation of the field φ under the generators of the symmetry.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 59 / 71 I More precisely, the last term is equal to

n Z X D −iq·x ∗ 0 0 0 d xe T 0 φ(x ) ... [j , φ(x )]δ(x x ) . . . φ(xn) 0 . − h | 1 i − i | i i=1

D−1 I Since the equal-time commutator is proportional to δ (~x x~i ), the previous expression becomes −

n X −iq·xi ∗ 0 0 e T 0 φ(x ) ... [Q, φ(x )]δ(x x ) . . . φ(xn) 0 . − h | 1 i − i | i i=1 where

δφ(x) = [Q, φ(x)]

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 60 / 71 I For a scale transformation: Z µ µν µ µ D−1 0 j = xνT ; ∂µ j = T ; = d x j , D D µ D D µ δφ(x) = [ , φ(x)] = i (d + x ∂µ) φ(x) , D where T µν is the energy-momentum tensor of the theory.

I We can neglect the ﬁrst term in the Ward identity by keeping terms up to (q0) (provided that there is no pole) and we can use the followingO equation

µ 2 µ 2 T (x) = v ∂ ξ(x); ∂µ j (x) = v ( ∂ ) ξ(x) µ − D − where v is related to the vev of the dilaton ﬁeld, denoted by ξ . h i I In particular scalar theories we ﬁnd:

D 2 v = − ξ 2 h i

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 61 / 71 I We introduce the LSZ operator:   h i n Z n Y D −ikj ·xj 2 2 LSZ i  lim d xj e ( ∂j + mj ) , ≡ k 2→−m2 − j=1 j j

2 2 where the limits kj mj put the external states on-shell, which has to be performed→ only − at the end.

I We apply it in the second term of the Ward identity: Z h i D −iq·x ∗ µ LSZ d x e T 0 ∂µj (x)φ(x ) . . . φ(xn) 0 h | D 1 | i n D (D) X = ( i) v (2π) δ ( k + q) + (q; k ,..., kn) , − j Tn 1 1 j=1

where we have Fourier transformed and extracted the poles of the correlation function to identify the amplitude + (a dilaton with Tn 1 momentum q and the other particles with momentum ki ).

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 62 / 71 I Performing the same operation with the last term of the Ward identity: ! h i n X −iq·xi ∗ LSZ e T 0 φ(x ) δφ(x ) φ(xn) 0 − h | 1 ··· i ··· | i i=1 n " X 2 2 µ ∂ = lim (ki + mi ) i d D (ki + q) µ − k 2→−m2 − − ∂k i=1 i i i D (D) Pn # (2π) δ j=1 kj + q (k ,..., k + q,..., k ) , 2 2 n 1 i n × (ki + q) + mi T

where all states j = i have already been amputated and put on-shell. 6

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 63 / 71 I The next step is to commute the differential operator passing the ith propagator and the δ-function, using the identity:

n  n  n X µ ∂ (D) X (D) X ki δ ( kj ) n(k1,..., kn) = δ ( kj ) ∂ki ν T i=1 j=1 j=1 n ! n−1 µν X µ ∂ X η + ki n(k1,..., kj ) . × − ∂ki ν T − i=1 j=1

I It is necessary to enforce momentum conservation whenever a derivative is acting on the amplitude. ¯ Pn−1 I We will denote this procedure for brevity by kn = k . − i=1 i

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 64 / 71 I Expanding n in the soft momentum q and following through with this procedure,T we ﬁnd:

n ( n X X ∂ i(2π)Dδ(D)( k + q) D n d k µ − j − − i ∂k µ j=1 i=1 i n ) X 2m2(k 2 + m2) ∂ i i i + µ lim 2 1 q µ − k 2→−m2 2 2 ∂k i=1 i i (ki + q) + mi i ¯ n(k ,..., kn) , × T 1 where we have used d = (D 2)/2, and neglected terms of (q1). − O

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 65 / 71 I We arrive at:

" n 2 n # X mi µ ∂ X µ ∂ v n+1(0; ki ) = 1 + q µ + D nd ki µ T − ki q ∂k − − ∂k i=1 i i=1 i n(k ) ×T i up to terms of order q0.

I One can repeat the same calculation with the current corresponding to the special conformal transformation:

µ µν 2 j = T (2xνxλ ηνλx ) (λ) − µ µ 2 ∂µ j = 2 xλ T = 2 v xλ( ∂ ) ξ(x) (λ) µ − 1 I The calculation gives the term of order q in the soft behavior.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 66 / 71 I The ﬁnal result is ( n 2 2 X mi µ ∂ 1 µ ν ∂ v n+1(q; k1,..., kn) = 1 + q µ + q q µ ν T − ki q ∂k 2 ∂k ∂k i=1 · i i i n X ∂ + D nd k µ − − i ∂k µ i=1 i

n 2 2 λ X 1 µ ∂ ∂ q 2 ki µ λ ki λ ν − 2 ∂k ∂k − ∂kiν∂k i=1 i i i ∂ ¯ 2 +d λ n(k1,..., kn) + (q ) . ∂ki T O

I In the gravity dilaton D nd 2 and there is no term in red appearing in the last line.− → I The knowledge of the subsubleading term in the soft dilaton has allowed to determine R∞ and then use the BCFW relations [Hui Luo and Congkao Wen (2016)].

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 67 / 71 Double-soft behavior

I We have used the Ward identities with two currents to compute the non-singular (regular) terms of the double-soft behavior. I The result is " n ! n ! 2 ¯ X ˆ X ˆ f T + (q , q , k ,..., kn) = D d + D D + D ξ n 2 1 2 1 − i i i=1 i=1 n n !# λ λ X ˆ X ˆ ¯ + (q + q ) K ,λ D d + D Tn(k ,..., kn) 1 2 ki − i 1 i=1 i=1 D 2 + (q2, q2, q q ); d − ; d = d + η O 1 2 1 2 ≡ 2 i i where fξ = v is the dilaton decay constant and

ˆ ˆ 1 2 Di = di + ki ∂k , Kk ,µ = kiµ∂ (ki ∂k )∂k ,µ di ∂k ,µ − · i i 2 ki − · i i − i I It is the same result that one obtains by performing two single-soft limits one after the other. [R. Marotta, M. Mojaza and PDV, JHEP 1709 (2017) 001.]

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 68 / 71 Conclusions and Outlook

I By imposing the conditions required by gauge invariance

µ ν q (Mµν f ηµν) = q (Mµν f ηµν) = 0 − − one can ﬁx the three leading terms in the soft behavior of both the graviton and the dilaton and the leading term of the Bµν. I In general, f can be an arbitrary function of the external momenta.

I This would prevent us to get a soft behavior for the dilaton.

I By performing explicit calculations in string theory, it turns out that f has a very simple form.

I Its presence has an important physical meaning when including open strings and discussing loops.

I This result, obtained by solving the previous relations, agrees with explicit string calculations apart of course from (bosonic+heterotic) string corrections to the term of order q1.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 69 / 71 I Those string corrections appear only in the soft behavior of the graviton.

I The soft behavior of the dilaton has no string corrections and it is the same in all string theories: it is universal.

I The string corrections are a direct consequence of the fact that the three-graviton vertex has string corrections with respect to the one of the Einstein-Hilbert action.

I The soft behavior of the dilaton contains the generators of scale and special conformal transformations.

I We have then considered a spontaneous broken conformal theory.

I We have shown that the Ward identities of scale and special conformal transformations ﬁx the three leading terms of the soft behavior of the Nambu-Goldstone boson, that we call ﬁeld theory dilaton.

I The soft behs. of the two dilatons are similar but not exactly equal.

I This comparison has been done at the tree level.

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 70 / 71 I Conformal invariance is, in general, explicitly broken by loop corrections that make the ﬁeld theory dilaton to acquire a mass proportional to the beta-function of the theory (used in Higgs as a dilaton).

I Soft behavior of the ﬁeld theory dilaton in = 4 super Yang-Mills that stays conformal invariant also at the quantumN level.

I On the other hand, the gravity dilaton stays massless in string perturbation theory and seems to have the same soft behavior at the loop level as in the tree diagrams.

I Why does the soft behavior of the string dilaton contain the generators of scale and special conformal transformations?

I Why is the function f so simple?

Paolo Di Vecchia (NBI+NO) Soft behavior Brussels, 17.05.2018 71 / 71