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,{ki,zi}) 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 ∗ ( , { , }) = ˆ( , )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) ∼ |zij| d z |z − zl| n+1 dΩ (z − z )(¯z − ¯z ) M i 0 ! ! ! j 2 2 X α q · km ¯zim zim ¯zmj |zij| 2 I ∼ ln Λ − ln |zij| + Li2 − Li2 − 2 ln ln + O(q ) i ¯ ¯ | | 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, {ki, zi}) = 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 ) = Sˆ(µν)(q, k ) + (q2) Mn+1 Mn ∗ q { i i} i Mn O n µ ν ˆ(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; εµν η = 0 . d µν Dilaton: Contracting with the projection tensor εµν = ηφ , n n n ρ m2 q·∂ Singular mass-terms: ˆ 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, {ki, zi}) = 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 |z − zi| 2 = i 2 = i i 1 i 1 i=1 All integrals involved (after partial expansion in q) Z Qn α0q·k j j ... |z − zl| 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) { i i} −1 0 0 1 1 O Can this be reproduce by a soft operator; Sˆµν = 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)|¯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). Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Subsubleading Order: Soft Theorem for the Graviton and Dilaton Obstruction: Presence of dilogarithms in the antisymmetric part Symmetric soft theorem through ssL: M(µν)( , ) = ˆ(µν) + α0ˆ(µν) M + O( 2) n+1 q ki SUniversal Sspin n q n µ ν ∂ ∂ ˆ(µν) 1 X ν µ µ ν µ ν ki ki ρ ρ Sspin = qσ ki ηρ + qρki ησ − ηρ ησ (ki · q) − qρqσ i + ¯i 2 k · q ∂ σ ∂¯ σ i=1 i i i ˆ(µν) 2 EFT origin of Sspin : Higher-order corrections to EH-gravity by L ∼ φR - term (vanishes in 4d) Z ( − √ 4 φ 1 d p µν 1 d−2 2 S = d x −G R − G ∂µφ∂ν φ − e (∂ B ) EFT κ2 [µ νρ] 2 d 12 − √ 2 φ ) 0 d−2 h 2 2 2 i 02 + α λ0 e Rµνρσ − 4Rµν + R + ··· + O(α ) 1 1 λ0 = 4 , 8 , 0: Bosonic, Heterotic, Type II Superstring [Zwiebach ‘85], [Metsaev, Tseytlin ‘87] µν M = , Nn+1 fixed by: qµ ν n+1 0 up to terms: µν [µνρ] [µρ],[νσ] {η , qρA , qρqσ B ,...} [Bern, Davies, Di Vecchia, Nohle ‘14], [Di Vecchia, Marotta, Nohle, M.M. ‘15], [IV], [Li, Lin, Zhang ‘18] 2 Assuming locality in q of Nn+1: Soft behavior completely fixed to O(q ) for the graviton. [Jackiw ‘68] Matin Mojaza (AEI-Potsdam) Soft Strings 10/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary The Universal Dilaton Soft Theorem Dilaton soft theorem universal for all string theories [II] φ 0ˆµν ηµν α Sspin = 0 n n n n φ µν X X ρ X qρqσ ρσ X q · q∂ ˆ ki ηµν SUniversal = 2 − Di + qρ Ki + SV,i − e (ki∂i ) 2q · ki q · ki i=1 i=1 i=1 i=1 Field Theory Consequence: tree φ ˆ 2 lim Mn×gravitons;1×φ = 0 = lim ηµν SUniversal Mn×gravitons + O(q ) α0→0 α0→0 Implies conformal invariance of graviton amplitudes! [Loebbert, Plefka, M.M. ‘18 ] Pn F.T. Pn µ F.T. (2 − i=1Di)Mn×gravitons = 0 , ( i=1Ki )Mn×gravitons = 0 (new) New relation explicitly checked to 6-pts! Origin of conformal generators remains a puzzle. Matin Mojaza (AEI-Potsdam) Soft Strings 11/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Subsubleading B-field Soft Behavior [µν] S1 contains the Bloch-Wigner (BW) dilogarithm (D2) 0 2 n h i [µν] α X 2 ρ [ν µ] ν [µ ρ] µ [ρ ν] zim S = i qρ kmk k + kmk k + kmk k D2 1 BW 2 3 i j i j i j zij i=6 j=6 m | {z } [µνρ] totally antisymmetric tensor: Ai [µνρ] I Gauge invariant by itself, qµqρAi = 0. I D cannot be expressed in terms of an operator acting on . 2 Mn Full soft behavior decomposes in separately gauge-invariant parts: [IV] (0) (1) B [µνρ] 2 (k ; q) = Sˆ + Sˆ (k ) + qρε (k ) + (q ) Mn;B i B B Mn i µν A i O n ˆ(1) X qρqσ ρ[µ ν]σ ¯ S = J J , (JL,i = Li + Σi, JR,i = Li + Σi) B k q L,i R,i i=1 i · This structure extends to all-orders in the soft expansion: (0) (1) B [µνρ] (k ; q) = Sˆ + S˜ (k + q) + qρε ˜ (k ; q) . Mn;B i B B Mn i µν A i All-orders “soft theorems” were derived for the graviton and photon in [Hamada, Shiu ‘18], [Li et al. ‘18] Matin Mojaza (AEI-Potsdam) Soft Strings 12/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary The Unified Massless Closed String Soft Theorem Graviton, dilaton and B-field soft behavior n h µν µν µν i [µνρ] o 2 =µ¯ν κ S + S + S + qρ ˜ (k ) + (q ) Mn;1 D −1 0 1 Mn A i O n µ X k kν Sµν = i i −1 k q i=1 i · n " ν µρ µ ¯ νρ # i X k qρ JL,i + Σi k qρ JR,i + Σi Sµν = i + i Σµν Σ¯ µν 0 − 2 k q k q − i − i i=1 i · i · n X qρqσ h i Sµν = Jρ(µJν)σ + Jρ[µJν]σ + α0Sˆ(µν) 1 k q i i L,i R,i spin i=1 i · In field theory: Full soft theorem for a ‘YM double-copy multiplet’. Matin Mojaza (AEI-Potsdam) Soft Strings 13/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary The Superstring Story I Observations [III, IV] a) = S = (S + S ) , Mn+1 Mn ∗ Mn ∗ bos. susy b) = b s , Mn Mn ∗ Mn No new types of integrals appear in S through (q2)! susy O µ ν S X ki ki 0 0 Sbos. = + O(q ) , Ssusy = 0 + O(q ) µν k · q i=1 i µ µ X k qρ νρ X k qρ νρ ∼ − S i Mb , ∼ − S i Ms Sbos. O(q0) i µν Ji n Ssusy O(q0) i µν Ji n q · ki q · ki ∼ (ˆ +α0ˆ )Mb , ∼ ˆ Ms + “( Mb )( Ms )”−α0ˆ Mb Sbos. O(q) S1 Sspin n Ssusy O(q) S1 n J n J n Sspin n I Soft theorem for a symmetric state in superstrings (µν) ˆ(µν) 2 = S 0 n + (q ) Mn+1 bos. α =0 M O Equivalent to field theory! But = Bosonic (and Heterotic) string. 6 Matin Mojaza (AEI-Potsdam) Soft Strings 14/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary The Multiloop Story [in progress] In the bosonic string it is possible to write a closed form expression for multiloop amplitudes using the N-point, h-loop, vertex operator. [Di Vecchia, Pezzella, Frau, Hornfeck, Lerda, Sciuto ‘88] Simplest case: One massless closed string +n tachyons [to appear] Z (h) µ ν n (0) (h)o M (ki; q, ¯ ) = dMh M + ( · ¯)N n;1 n;1 G0→Gh n;1 By only using general properties of Gh we are able to calculate the soft function for the graviton and (possibly) the dilaton through O(q2) for any h ! ∂z∂¯zGh(z, w) = πδ(z − w) − T (z) Z 2 d z∂z∂¯zGh(z, w) = 0 Σh 0 exp α kiqGh(zi, zi) = 0 Matin Mojaza (AEI-Potsdam) Soft Strings 15/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary New Multiloop Results Soft graviton theorem: Unchanged ! But infrared divergent Soft dilaton theorem: Modified in an important way ! " # n n n m2 (h) µν X X µ X i q·∂k (h) 2 (ki; q, η ) = 2 i + h(D 2)+ qµ e i n + (q ) Mn;φ φ − D − Ki − k q M O i=1 i=1 i=1 i · Scaling property " n # X (h) D 2 ∂ 0 ∂ (h) 2 i + h(D 2) n = − gs √α n − D − M 2 ∂g − √ 0 M i=1 s ∂ α Thus " n n 2 # D 2 ∂ ∂ X µ X m q·∂ √ 0 i ki 2 n;φ = − gs α + qµ i e n + (q ) M 2 ∂gs − ∂√α0 K − k q M O i=1 i=1 i · ∞ X (h) M ≡ M h=0 Remark D 2 ∂ 0 ∂ − gs √α κD = 0 2 ∂gs − ∂√α0 Matin Mojaza (AEI-Potsdam) Soft Strings 16/17 Introduction Soft Calculations Soft Theorems Unification, Superstrings & Multiloops Summary Summary & Some Open Questions µν I Studied soft behavior of Mn+1 generically in different string theories (µν) [µν] I Found ssL soft theorem for Mn+1 and sL soft theorem for Mn+1 Holomorphic soft theorem at sLO Appearance of α0 at the ssLO for the graviton I Gauss-Bonnet term modifies graviton soft theorem at ssLO Graviton soft theorem is different in bosonic/heterotic/superstring I The dilaton soft theorem contains two surprises: 1. It is universal among all string and field theories! 2. Contains the generators of dilatation and special conformal transformation! I These results seems to be extendable to all-orders in loops Up to IR divergences I Is there a symmetry reason behind the universality of the soft dilaton? The dilaton does not possess a (known) ‘gauge’ symmetry I Is there an analog of BMS symmetry for the Yang-Mills double copy? Matin Mojaza (AEI-Potsdam) Soft Strings 17/17