Massive strings from a haunted field theory

In this work we present the $\alpha'$-exact background equations of motion of the bosonic chiral string (also known as Hohm-Siegel-Zwiebach model), with the spin two ghost fields integrated out. This is the first instance of a worldsheet model in which all corrections are fully determined in a generic curved spacetime. As a concrete cross-check, we find complete agreement between all three-point and a sample of four-point tree level scattering amplitudes computed using field theory methods and the chiral string prescription. These equations of motion provide a field theoretical shortcut to compute worldsheet correlators in conventional bosonic strings (with arbitrary number of massless and mass level one states), and outline a new perspective on massive resonances in string theory.

In this work we present the α ′ -exact background equations of motion of the bosonic chiral string (also known as Hohm-Siegel-Zwiebach model), with the spin two ghost fields integrated out.This is the first instance of a worldsheet model in which all corrections are fully determined in a generic curved spacetime.As a concrete cross-check, we find complete agreement between all three-point and a sample of four-point tree level scattering amplitudes computed using field theory methods and the chiral string prescription.These equations of motion provide a field theoretical shortcut to compute worldsheet correlators in conventional bosonic strings (with arbitrary number of massless and mass level one states), and outline a new perspective on massive resonances in string theory.

I. OVERVIEW
Many deep and unexpected connections in physics were uncovered due to the fruitful, though not often evident, interchange between strings and fields.
An explicit example is the fact that some scattering amplitudes in quantum field theory have a dual description in terms of integrations over two-dimensional (Riemann) surfaces with marked points.This is the basic idea underlying the Cachazo-He-Yuan amplitudes (CHY) [1], flagrantly reminiscent of string theory and its worldsheet.However, conventional string theory has an infinite physical spectrum and cannot be described through ordinary field theory methods.There is no finite Lagrangian for such an enterprise.
When focusing on massless theories, CHY amplitudes admit a more fundamental, string-like description in terms of ambitwistor strings [2].In spite of living in two dimensions, ambitwistor strings are chiral theories.Their constituents are function of only one of the light-cone directions of the worldsheet.In addition, ambitwistor strings have no dimensionful parameter, hence a massless spectrum.Indeed, they are related to a chiral model proposed by Hohm, Siegel, and Zwiebach (HSZ) [3] through a tensionless limit [4][5][6][7][8].
The HSZ model is supposed to be the worldsheet theory underlying an effective spacetime description of the massless string spectrum, with manifest T-duality.It was argued to be a consistent truncation of string theory with complete control over the corrections in α ′ (the string length squared) [3,9,10].As shown in [4] (see also [11] for further details), this model may be derived from the first order Polyakov action through a singular gauge choice, with a degenerate worldsheet metric.In this sense, chiral strings are more than a particle but less than a string.Though containing ghosts, their physical spectrum is finite, suggesting a possible Lagrangian description.
The heterotic HSZ model has been recently used to determine conformal field theory (CFT) correlators in conventional string theory amplitudes [12].Those were tree level results, involving an arbitrary number of massless fields and only one state from the first massive level.More intriguingly, they were derived using a mix of string theory techniques and a field theory involving ghosts.In hindsight, the appearance of ghosts should not come as a surprise.The impressive consis-tency of string theory relies on an infinite tower of massive states.But when trying to describe specific mass levels through a Lagrangian the emergence of a "sick" field theory may be expected.
The breakthrough of [12] was at the time limited by two factors.First, a lack of control of the gravitational sector of the theory.Second, the unique massive multiplet of the chiral string, matching the degrees of freedom of the first mass level of the open superstring.While higher mass levels may now be individually accessed through the asymetrically twisted strings [13], their field theory realization is more involved, naturally containing higher spins.The bosonic chiral string, on the other hand, offers the possibility of the exact computation of its tree level dynamics.That includes the usual massless bosonic string spectrum (graviton, Kalb-Ramond two-form, dilaton) and two spin-two massive states -the ghosts of the HSZ model.This possibility stems from the fact that the worldsheet action is free while the coupling to the background is exclusively governed by the BRST charge.Its nilpotency should yield the field equations describing the spacetime dynamics of the physical spectrum.
In this work we present for the first time the exact equations of motion of the HSZ model (bosonic chiral string).The respective field theory has a six derivative kinetic term after integrating out the auxiliary spin-two fields, including Chern-Simons-like corrections for the Kalb-Ramond field-strength [14,15] and the complete expressions for the α ′ corrections.In analogy with the results of [12], we provide a field theoretical shortcut to obtain the relevant CFT correlators for computing N -point tree level (open and closed) bosonic string amplitudes with an arbitrary number of massless and mass level one states.This is a concrete step to understand the role of mass in string theory from a novel perspective.One in which mass levels are singled out, with a dual description as ghosts in a gravitational theory.

II. BOSONIC CHIRAL STRINGS
We start with an alternative description of the HSZ model.It can be viewed as an ambitwistor string in which one of the states of the zero momentum cohomology acquires a vacuum expectation value.In order to see this, consider the ambitwistor gauge fixed ac-tion and energy-momentum tensor [2], where ∂ ≡ ∂/∂z, ∂ ≡ ∂/∂ z.The target space coordinates are denoted by X m , with canonical conjugate P m .The ghost pairs {b, c} and { b, c} are respectively associated with the generators T and H = − 1 2 η mn P m P n , where η mn is the flat space metric.The BRST charge is simply Q 0 = ¸(cT − bc∂c + cH), with {Q 0 , b} = T and {Q 0 , b} = H.Requiring its nilpotency fixes the spacetime dimension to be d = 26.
Physical states are defined to be in the ghost number two cohomology of Q 0 , annihilated by the zero mode of the b ghost [16].In particular, there are two zero-momentum states, with relative dimension quartic in length.The first generates constant deformations of the flat space metric η mn → η mn + h mn , as it couples to the BRST current as [b −1 , U h ] = ch mn P m P n .Similarly, the coupling of U t is given by It generates a tensile deformation that will be parametrized by (α ′ ) 2 , becoming the only dimensionful parameter of the model.In this case, the new BRST charge may be expressed as where The gauge algebra of T and H is given by It can be seen as two copies of a chiral Virasoro algebra with generators T ± = (T ± α ′ H)/2, and gives rise to a sectorized interpretation of the chiral string [11].
The physical states of the chiral model have been discussed in detail in [17] (in the second order formalism, they can be nicely described with the usual oscillator construction [18,19]).The massless level is the conventional bosonic string spectrum (graviton, Kalb-Ramond two-form, dilaton), and there are two spin 2 states with mass m 2 = ±4/α ′ .The precise form of the vertex operator U is not immediately relevant, but we are interested in the corresponding deformation of the BRST charge.Up to BRST-exact contributions and total derivatives, it can be generically expressed as The fields {A m , B m n , C m , D, g mn , gmn } will then describe the background dynamics.

III. CURVED SPACETIME
In the bosonic chiral string, the coupling to the curved background is determined solely in terms of deformations of the BRST charge.Motivated by ( 9), our idea is to propose modified operators T and H that still satisfy the algebra ( 8).This will ensure the nilpotency of the BRST charge ( 6) while imposing consistency conditions -the field equations -for the background.At a semiclassical level, a similar procedure has been long known in conventional (super) string theory [20][21][22][23].More recently, it has been extended to ambitwistor strings in NS-NS backgrounds [24].
The new operators are defined as The spacetime metric is g mn (with inverse g mn ).For later convenience we decompose B m n as with S mn and b mn symmetric and antisymmetric in m ↔ n, respectively and without loss of generality.
There are slightly more general deformations of H that spoil the T − H algebra but still lead to a nilpotent BRST charge.However, they correspond to pure gauge degrees of freedom in the field theory setup.Towards (8), first notice that the OPE (8a) is left unchanged using (10).Next, we have to impose that the new H is a primary conformal operator, cf.equation (8b).A direct computation leads to (13) with α m = χ m = 0 provide algebraic solutions for A m and C m , while β = 0 yields a scalar equation of motion.Finally, the OPE of H in (11) with itself, c.f. (8c), can be computed to be The ellipsis denote other poles in the OPE that are not independent from {α m , β, β m n , β G mn , χ m , Σ, Σ mn }.For instance, there is a cubic pole with numerator ∂Σ/2.
The coefficient Σ mn can be expressed as implying that S mn is solved in terms of g mn and C m .Then, after defining the dilaton field as where g = det(g mn ), equation ( 15) becomes Here ≡ g mn ∇ m ∇ n , with covariant derivative ∇ m .α ′ corrections to the dilaton equation of motion appear through the scalar curvature R.
The coefficient β m n , given by may also be expressed as a symmetric and an antisymmetric piece.The symmetric part is the algebraic solution for gmn , which takes the form This solution may then be replaced in Σ, At leading order in α ′ , the vanishing of Σ fixes the spacetime dimension to be 26.The next order vanishes on the support of the other algebraic solutions.
The antisymmetric part of ( 21) does not depend on gmn , and can be neatly cast as The Kalb-Ramond field-strength, H mnp , includes a gravitational Chern-Simons correction, Ω mnp , The Christoffel symbols, Γ p mn , and the Riemann tensor, R q pmn , have the usual definition, with Ricci tensor R mn = R p mpn and scalar R = g mn R mn .The α ′ correction of the field-strength ( 25) is reminiscent of Lorentz group Chern-Simons form found by Green and Schwarz [14].It had already been identified in the HSZ model [15].In more practical terms, b mn is a pseudo-tensor that receives α ′ corrections in the diffeomorphism transformations.H mnp , on the other hand, is the fully covariant object.
Finally, β G mn = 0 can be written as with It is clear from (29) the auxiliar character of the massive spin 2 states.They can be exactly integrated out of the system, effectively inducing a higher derivative gravitational theory.This is a signature feature of the bosonic and heterotic chiral strings (see also [17]).Their massive spectrum consists of ghosts states with the same degrees of freedom of the first mass level in the corresponding open string.

IV. SCATTERING FIELDS AND STRINGS
In conventional string theory, the background equations of motion emerge from the Weyl anomaly in the worldsheet, order by order in α ′ .Here, on the other hand, equations ( 20), (24), and ( 29) account for all α ′ corrections, completely governing the classical dynamics of the system.They can be used to determine any tree level scattering amplitude involving the physical states (massless and massive).Alternatively, these amplitudes may be computed directly from the chiral model [11] or using twisted strings [4,25].
This interplay between (twisted) strings and fields was explored in [12] in order to determine CFT correlators using a purely field-theoretical input through perturbiner methods [26][27][28][29][30].In the present case, this input is slightly more involved because it contains (1) gravity, and (2) a higher derivative kinetic term due to the spin 2 ghosts.Fortunately, the perturbiner was extended to gravitational theories coupled to matter in [31] (and later in [32], in the context of doube field theory).The higher derivative kinetic term is but a small detour of the traditional (quadratic) case.
A necessary step in working with the perturbiner is to fix the gauge symmetries of the field theory.We will work here with fixing the form symmetry and the spacetime diffeomorphisms.The latter, which we dub dilaton-de Donder gauge, is a convenient choice for our string frame equations of motion.As opposed to the Einstein frame, with metric g mn = e −φ/6 g mn , the string frame intermingles the dilaton and the graviton.Indeed, the linearized solutions of ( 20), (24), and (29), i.e. singleparticle states, can be parametrized as bmn (x) = b mn e ik b •x , (32a) where (k ± ) 2 = ∓4/α ′ , and k 2 = 0 otherwise.The polarizations denote the dilaton, φ, the Kalb-Ramond 2-form, b mn , the graviton, g 0 mn , and the ghosts, g ± mn .They are traceless and transversal with the respective momenta, and there is a residual gauge symmetry δb mn = k m γ n − k n γ m , and Next, we define the multiparticle expansions: The sums are over all ordered words J = j 1 j 2 . . .j n (j 1 < ... < j n ) composed of single-particle labels j (letters).The multiparticle momenta are given by k J = k j1 +...+k jn .Multiparticle currents are denoted by B Jmn , G Jmn , and Φ J , and reduce to single-particle polarizations for one-letter words J = j.See [31] for notation.
In order to satisfy the classical equations of motion, the expansions (33) lead to recursive definitions of the multiparticle currents of the form with s J = k 2 J denoting the generalized Mandelstam variables.The interaction terms on the right hand side are built out of powers of multiparticle currents with subwords of J.For instance, where the sum is over all possible ways of partitioning the ordered word J into two ordered subwords K and L, such that K ∪ L = J.The ellipsis also involve splittings with more than two subwords.The precise forms of {B int Jmn , G int Jmn , Φ int J } are lengthy but straightforward to determine via the equations of motion.More details will be given in [33].
We are now ready to compute the tree level scattering amplitudes of the model, the final step in the perturbiner method.They are defined as ..N mn ), (36c) on the support of momentum conservation, k 1...N = 0.For example, the three-point amplitude involving the dilaton and two Kalb-Ramond forms, using either (36a) or (36c), is given by The residual gauge symmetries of the single-particle polarizations b mn and g 0 mn leave the amplitudes invariant in the gauge (31), which translates to multiparticle currents as As the ultimate consistency check of the field equations (20), (24), and ( 29), we should compare the amplitudes (36) with the ones computed using the chiral string CFT.There, the unintegrated vertex operators describing the physical states take the form where P m ± = (P m ± 1 α ′ ∂X m ).In the dilaton vertex, ηmn = η mn − k m q n − k n q m , where q m is a reference vector satisfying q • k = 1 and q 2 = 0. Now, using the results of [11], we were able to cross check all threepoint and an assortment of four-point tree level amplitudes of the bosonic chiral string (see also [25,34]) and the output described above.For example, the chiral string analogous of ( 37) is given by where N depends on the CFT data and vertex normalizations.An overall numerical factor cannot be fixed using the perturbiner method, so this is the only extra input from the chiral string.Once fixed, all tree level amplitudes match exactly.

V. SUMMARY & PROSPECTS
In this work we have derived the α ′ -exact background field equations of the HSZ model (bosonic chiral string), i.e. equations (20), (24), and (29).Up to our knowledge, this is the first example of a string model in which all α ′ corrections have been completely determined for a generic background.
In order to support our results, we performed a thorough cross check between tree level amplitudes computed using two fundamentally different approaches.Namely, the usual string prescription and the field-theoretical perturbiner method.Furthermore, the background field equations have the expected α ′ → ∞ and α ′ → 0 limits.The former is the bosonic ambitwistor string, corresponding to a fully covariantized version of [16].The latter is given by the conventional bosonic string (see e.g.[35]).We note, in particular, that the α ′ → 0 limit is obscured in the worldsheet action and the BRST charge.This suggests that a different set of variables might be more suitable to effectively built a worldsheet theory for (super) gravity.For finite α ′ , two states emerge, with opposite mass squared.Indeed, the model is α ′ → −α ′ symmetric.Unlike in massive bigravity [36,37], these spin 2 states have an auxiliary role and can be integrated out of the equations of motion, effectively introducing a higher derivative kinetic term with ghosts (see also [38] and references therein for other results on massive spin 2 fields in string theory).A separate paper is being prepared with an extended, more detailed presentation of the results introduced here [33].It would still be interesting to investigate whether equations ( 20), (24), and ( 29) have nontrivial solutions (such as ghost condensates [39], also suggested in [10]) and, if so, their classical properties.Additionally, one could also investigate if and how T-duality is manifested in these equations (see also [40] for a related discussion).In the chiral formulation, it is not clear how to identify, for instance, the winding modes of X m , though this can be easily done in the second order formulation [41].
The HSZ model was considered the worldsheet theory behind the so-called doubled α ′ geometry [3], which has been ambitiously developed in several directions (to cite a few [42][43][44][45]).Our results directly clash with different ideas related to double α ′ theories.Spectrum wise, and as expected [4,17,25], we observe the usual massless spectrum of the closed string plus two massive spin two states.This is confirmed by the analysis of physical poles of the amplitudes.There are no massive scalars, unlike in [10].Relatedly, the massive fields can be fully integrated out and lead to a finite series of higher-derivative corrections (i.e. a finite number of α ′ corrections).The background field equations we obtain have no (α ′ ) 3 corrections or higher.Barring some intriguing similarities, such as Green-Schwarz like corrections to the Kalb-Ramond field strength, and involving up to six derivatives, the doubled α ′ geometry and the bosonic chiral string lead to different spacetime theories.Therefore, unless further constraints are imposed, the HSZ model cannot be seen as a consistent truncation of string theory.
Besides providing a more complete picture of the bosonic chiral string, we can mimic the construction of [12] in order to convert field theory amplitudes (36) to their respective chiral string amplitudes before the moduli space integrations.The reasoning is more or less straightforward.First, we have to remember that the chiral worldsheet is equivalent to a second order worldsheet with target space coordinates satisfying [4] X m (z, z)X n (y, ȳ) ∼ α ′ 2 η mn [ln(z− ȳ)−ln(z−y)], (42) with a sign flip between the holomorphic and antiholomorphic sectors.The CFT correlators factorize into a chiral and an antichiral piece, each matching a given open-string-like configuration involving massless vectors and massive spin 2 states.Since the chiral correlators are universal, the amplitudes computed via (36) can be effectively translated to ordinary open and closed string scattering at tree level.The method is analogous to the one described in [12], but now involving a colorless theory.The bosonic case is particularly interesting, since the relevant CFT correlators involving massive states are more tedious to compute.It should, nonetheless, be extended to the more interesting ones of the spinning string.They involve additional ingredients such as picture changing, spin fields, and we plan to address this in a future work.