Soft Strings The Unified Soft Behavior of the Graviton, Dilaton and B-field in any String Theory Matin Mojaza Albert-Einstein-Institut Max-Planck-Institut fur¨ Gravitationsphysik Potsdam, Germany July 12, 2018 New Frontiers in String Theory 2018 Yukawa Institute for Theoretical Physics, Kyoto University Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary (Spontaneously Broken) Symmetries in Observables Soft emission theorems: universal low-energy properties of amplitudes Factorization only symmetry-dependent (universal) −1 0 1 p S(τ q; ki) = τ SL + τ SsL + τ SssL + ··· + O(τ ) Famous Examples - Recent revival ‘58 Low’s soft photon theorem - gauge invariance SL; SsL [‘15] WI of asymptotic 2D Kac-Moody symmetry - Strominger et al. ... tree tree ‘64 Weinberg’s soft graviton theorem - gauge invariance SL; (SsL ; SssL ) [‘14] WI of asymptotic BMS symmetry - Strominger et al. ... (Cachazo, Strominger) ‘65 Adler’s pion zero condition - shift symmetry SL; SsL New EFTs from soft theorems and Soft Bootstrap - Cheung et al., Elvang et al, ... ‘66 Double-soft pion theorem - coset symmetry SL; SsL Many new double-soft theorems - field theory, string theory, etc. Matin Mojaza (AEI-Potsdam) Soft Strings 2/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary This Talk: Scattering of soft (massless, closed) strings Soft emission of gravitons, dilatons and Kalb-Ramond in string theories Work in collaboration with Paolo Di Vecchia and Raffaele Marotta: [I] JHEP 1505 [arXiv:1502.05258] [IV] JHEP 1710 [arXiv:1706.02961] [II] JHEP 1606 [arXiv:1604.03355] [V] JHEP ???? [arXiv:180X.XXXX] [III] JHEP 1612 [arXiv:1610.03481] Main results: I String corrections to new graviton soft theorem, I New dilaton soft theorem, I New Kalb-Ramond soft theorem I A unified operator for them all (through subleading order) I New multiloop results using the string multiloop construction Related work on soft strings: [Ademollo et al. ‘75], [Shapiro ‘75], [Schwab ‘14], [Avery, Schwab ‘15], [Bianchi, Guerrieri, ‘15], [Sen; Sen, Laddha ‘17-‘18], [Higuchi, Kawai ‘18] + many other recent QFT soft theorem papers. Matin Mojaza (AEI-Potsdam) Soft Strings 3/17 6.2 Closed String Amplitudes at Tree Level Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary The tree-level scattering amplitude is given by the correlation function of the 2d theory, evaluated on the sphere, m (m) 1 1 SString Amplitudes and their soft limits = X ge− Poly V (p ) A g2 Vol D D Λi i s i=1 ! " where VΛi (pi)arethevertexoperatorsassociatedtothestates. We want to integrate over all metrics on the sphere. Tree-level string amplitude At first glance that sounds rather daunting but, of course, we have the gauge symmetries of diffeo- Qn morphisms and Weyl transformations at our dis- R i=1 dzi n(k1;:::; kn) V1(z1; k1) Vn(zn; kn) [ c:c:]closed posal. Any metric on the sphere is conformally M ∼ dΩM h ··· i ⊗ equivalent to the flat metric on the plane. For ex- ample, the round metric on the sphere of radius R Figure 35: can be written as Key observation (bosonic sector) [I] 4R2 ds2 = dzdz¯ (1 + z 2)2 Z Qn Z n | | i=1 dzi Y n+1(q; k1;:::; kn) V1(z1; k1) Vn(zn; kn) dz Vq(z; q)Vj(zj; kj) which is manifestly conformally equivalent to the plane, suMpplemented by the point at ∼ dΩ h ··· i h i infinity. The conformal map from the sphere to the plane is the stereographic projection M j=1 depicted in the diagram. The south pole of the sphere is mappedtotheorigin;the | {z } M (k ;:::;k ) | {z } north pole is mapped to the point at infinity. Therefore, instead of integrating over all n 1 n Sq(q;fki;zig) metrics, we may gauge fix diffeomorphisms and Weyl transformations to leave ourselves with the seemingly easier task of computing correlation functions on the plane. Follows when Vi can be exponentiated. 6.2.1 Remnant Gauge Symmetry: SL(2,C) There’s a subtlety. And it’s a subtlety that we’ve seenExistence before: there isof a residual a soft theorem for q ki implies gauge symmetry. It is the conformal group, arising from diffeomorphisms which can be undone by Weyl transformations. As we saw in Section 4, there are an infinite number M ∗ ( ; f ; g) = ^( ; )M + O( p) of such conformal transformations. It looks like we have a whole lot of gauge fixing n Sq q ki zi S q ki n q still to do. However, global issues actually mean that there’s less remnExtendableant gauge symmetry to than multi-soft expansions you might think. In Section 4, we only looked at infinitesimal conformal transforma- tions, generated by the Virasoro operators L , n Z.WedidnotexaminewhetherFor double-soft gluons and scalars, see [1507.00938, P. Di Vecchia, R. Marotta, M.M.] n ∈ these transformations are well-defined and invertible over all of space. Let’s take a Matin Mojaza (AEI-Potsdam) Soft Strings 4/17 –130– Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Polarization stripped amplitude of a soft massless (NS-NS) closed state I Amplitude linear in polarization vectors: µν Mn+1(q; ki) = q,µ¯q,ν Mn+1(q; ki) I Physical polarizations (KLT: open×open = closed) µ ν (µ ν) µν µν [µ ν] q ¯q = q ¯q − ηφ q · ¯q + ηφ q · ¯q + q ¯q | {z } | {zµν } µν | {zµν } φ g B with µν µ ν ν µ µν η q ¯q q ¯q 2 2 p η = − − ; q ¯q = 1 ; q = ¯q = 0 ; q ¯q = D 2 φ D 2 · · − − I On-shell gauge invariance implies (q · q = 0) µν ν µν µν qµMn+1(q; ki) = q f (q; ki) ) qµ(Mn+1(q; ki) − η f (q; ki)) = 0 ηµν irrelevant for the graviton and B-field, but not for the dilaton! Matin Mojaza (AEI-Potsdam) Soft Strings 5/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Soft Massless Closed Bosonic String graviton, dilaton, Kalb-Ramond Simplest case: Bosonic string scattering on n closed tachyons (zij ≡ zi − zj) Z Q 2 n Z n n µ ν d zi α0 · 0 k kj µν i Y ki kj 2 Y α q·kl X i M (q; ki) ∼ jzijj d z jz − zlj n+1 dΩ (z − z )(¯z − ¯z ) M i<j l=1 i;j=1 i j | {z } | {z } M (k ;:::;k ) µν µ j n 1 n S (q;fk ;z g)≡P k kν I q i i i;j i j i j Soft-expansion of (master-integral) Leading term independent of z - Weinberg Soft Theorem Ii i 0 0 2 1 i 2 0 X (α ) X X I ∼ @1 + α (kjq) ln jzijj + (kjq)(klq) ln jzijj ln jziljA i α0 · q ki j6=i 2 j6=i l6=i 0 X 2 2 2 + α (kjq) ln jzijj + ln Λ + O(q ) j6=i 0 ! ! ! j 2 2 X α q · km ¯zim zim ¯zmj jzijj 2 I ∼ ln Λ − ln jzijj + Li2 − Li2 − 2 ln ln + O(q ) i ¯ ¯ j j m6=i;j 2 zij zij zij zim + ([µ ν]) 1 µ ν ν µ Decomposed soft function is physical and cutoff-free: v w = 2 (v w − v w ) µν X (µ ν) j i X [µ ν] j i Sq (q; fki; zig) = ki kj (Ii + Ij ) + ki kj (Ii − Ij ) i;j | {z } i;j | {z } Dilogs vanish Dilogs: Bloch-Wigner [µν] Here: antisymmetric part integrates to zero, i.e. S = 0, consistent with parity. Mn ∗ q Matin Mojaza (AEI-Potsdam) Soft Strings 6/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Soft graviton or dilaton emission from n-tachyon interaction For the symmetrically polarized part we find [I, IV] (µν) = S(µν)(q; k ; z ) = S^(µν)(q; k ) + (q2) Mn+1 Mn ∗ q f i ig i Mn O n µ ν ^(µν) X ki ki qρ (µ ν)ρ 1 qρqσ µρ νσ S (q; ki) = i k J : J J : q k − q k i i − 2 q k i i i=1 · i · i · i µν µν µν ¯ µν [µ ν] [µ ν] [µ ν] J = L +Σ + Σ = 2i k @k + @ + ¯ @¯ No soft α0-operator. :: normal ordering g µν Graviton: Consistent with field theory results; "µν η = 0 : d µν Dilaton: Contracting with the projection tensor "µν = ηφ , n n n ρ m2 q·@ Singular mass-terms: ^ P P P i ki Sdilaton = 2 i + qρ i q·k e − i=1 D i=1 K − i=1 i ρ 1 ρ 2 ρ ρµ Di = ki · @k and K = k @ − (ki@k )@ −iΣ @k ,µ i i 2 i ki i ki i i dilatation and special conformal transformation ! Matin Mojaza (AEI-Potsdam) Soft Strings 7/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary One soft + n hard massless closed strings Soft part of µν = Sµν Mn+1 Mn ∗ µ µ ¯ ν Z n θ q k n θ ¯ q ν µν 2 P i i α0 i P j j α0 ki S (q; fki; zig) = d z + + (z−z )2 2 z−z (¯z−¯z )2 2 ¯z−¯z i=1 i i j=1 j i " r # " r # n α0 n α0 n ¯ 0 P θii·q P θi¯i·q Y α q·ki × exp − z−z exp − ¯z−¯z jz − zij 2 = i 2 = i i 1 i 1 i=1 All integrals involved (after partial expansion in q) Z Qn α0q·k j j ::: jz − zlj l I 1 2 = d2z l=1 i1i2::: ( − )( − ) ··· (¯ − ¯ )(¯ − ¯ ) ··· z zi1 z zi2 z zj1 z zj2 can be related to the master integral by simple tricks as: j 0 −1 @ Ii −I j α q·ki j ij i i IBP: Iii = 1 − 2 Ii and PF: Ii = @zi ¯zi − z¯j The very lengthy end result schematically reads: [II, IV] Sµν (τq; k ; z ) = τ −1S(µν) + S(µν) + S[µν] + τ S(µν) + S[µν] + (τ 2) f i ig −1 0 0 1 1 O Can this be reproduce by a soft operator; S^µν = Sµν ? Mn Mn ∗ Matin Mojaza (AEI-Potsdam) Soft Strings 8/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Leading and Subleading Orders: A Holomorphic Soft Operator µν V (z; k) = Vµ (z; k) V¯ ν (¯z; ¯k) closed open × open ¯k=k Leading soft terms: just a function of momenta and manifestly symmetric: n µ ν µν X k k S = i i −1 k · q i=1 i Subleading soft terms: All reproduced by a holomorphic soft theorem: [I,IV] n " ¯ν µρ µ ¯ νρ # q k (L + Σ) qρk (L + Σ)¯ µν X ρ i i ¯ Mn+1 = −i + Mn(ki; i; ki; ¯i) O(q0) k · q ¯ · ¯k=k i=1 i ki q µν ¯µν ¯ µν Symmetric part: (L + L )Mn(k; k)j¯k=k = L Mn(k) implies: n (µ ν)ρ (µν) X qρk J S = −i i ; ( J ≡ L + Σ + Σ)¯ 0 k · q i=1 i B B Antisymmetric part: Gauge invariance, q µν ! q µν + qµχν − qν χµ, implies n [ν ( − ¯ )µ]ρ n X qρki Li Li ¯ X ¯ µν ¯ Mn(ki; i; ki; ¯i) = (Σi − Σi) Mn(ki; i; ki; ¯i) ; k · q k=¯k k=¯k i=1 i i=1 Antisymmetric soft theorem (consistent with zero for tachyons): n " [ν # [µν] X ki qρ ¯ µ]ρ 1 ¯ µν M = −i (Σi − Σi) − (Σi − Σi) Mn(ki; i; ¯i) + O(q) n+1 q · k 2 i=1 i Matin MojazaAt (AEI-Potsdam) ssLO: obstruction by the Bloch-Wigner-Dilog,Soft Strings but the soft theorem leads to a 9/17 universal decomposition (to appear).