aa r X i v : . [ h e p - t h ] F e b Chiral Superstring and CHY Amplitude
February 23, 2017 YITP-SB-17-10
Yuqi Li a ∗ , Warren Siegel b † a Department of Physics and AstronomyState University of New York, Stony Brook, NY 11794-3840 b C. N. Yang Institute for Theoretical PhysicsState University of New York, Stony Brook, NY 11794-3840
Abstract
We calculate the chiral string amplitude in pure spinor formalism and takefour point amplitude as an example. The method could be easily generalized to N point amplitude by complicated calculations. By doing the usual calculationsof string theory first and using a special singular gauge limit, we produce theamplitude with the integral over Dirac δ -functions. The Bosonic part of theamplitude matches the CHY amplitude and the Fermionic part gives us thesupersymmetric generalization of CHY amplitude. Finally, we also check thedependence on boundary condition for heterotic chiral string amplitudes. ∗ [email protected] † [email protected]://insti.physics.sunysb.edu/ ∼ siegel/plan.html Introduction
A few years earlier, Cachazo, He and Yuan (CHY) [1–4] introduced one kind of string-like prescription of tree-level amplitudes of N massless particles, in which they introducedone integral over only the z -dependent part and fixed the locations of each external linecorresponding to N particles by solutions of a series of formulas inside the δ -functions.Those N − δ -functions are called “Scattering Equations” by CHY,which have ( N − δ -functions (see, e.g., [6, 7]). Shortly after, Mason and Skinner(MS) [5] introduced one kind of string amplitude with a δ -function inserted into the vertexoperators of their ambitwistor string theory. By taking the infinite-tension limit ( α ′ → δ -functions were still inserted inthe vertex operators.Soon after, Siegel [9] introduced an approach closely related to the standard string theoryby treating the singular worldsheet gauge (HSZ gauge [10]) as a singular limit with somesimple modifications of the boundary conditions. Therefore, the integration over ¯ z withstandard vertex operators under the HSZ gauge limit produces the CHY δ -functions; thatis, if we take the rules that first evaluate the amplitude in the usual conformal gauge andsubstitute the propagators with singular HSZ gauge limit before integrating over ¯ z , theresults after integration over ¯ z only depend on the z coordinate with the correct number,namely N −
3, of Dirac δ -functions inserted in the integrand leaving the boundary conditionsstill simply modified. Furthermore, if we keep the modified boundary conditions and do thecalculations in the usual conformal gauge, the scattering-equation δ -functions are not seenexplicitly, and the results are the same since they should be gauge independent [11].Recently, we are motivated by the fact that ¯ δ ( k · P ) insertion of MS ambitwistor stringprescription could be treated as the (generalized) picture-changing operator; then, afterBRST transformation, the calculation could be changed into formulas with the usual ¯ z dependence in X . Noticed that we use the integrated version in our calculations (such as R P , c = R c and b = R b ) instead of the unintegrated version, the ambitwistor stringmethod [12] would be BRST equivalent to the calculations in the normal string theory withsome additional gauge limits, and the additional gauge limits lead to the localization of theexternal lines on the Riemann surface we integrated on by δ -functions. The similar discussioncould also be found in [13] and our upcoming paper of loop-level calculations.In this paper, we first discuss the gauge dependence of different prescriptions and calculatethe four-point amplitude of massless states using the pure spinor formalism of HeteroticStrings with the singular HSZ gauge. Namely, we do the usual operator product expansions(OPEs) of four massless states under the pure spinor formalism and then substitute thesingular limit of ¯ z . The ¯ z integration of the Koba-Nielsen factor produces the CHY δ -function and the current algebra part produces the Parke-Taylor-like factor by directly takingthe singular gauge limit. The four-Boson part of the amplitude matches the usual Yang- Unlike usual CFT in string theory, we start with unintegrated version of vertex operator with ghostinserted, and then get the integrated N − z , the CHY amplitude of four gravitons are produced with alsothe similar supersymmetric generalization as in the SYM case. Furthermore, the flip-signmethod of the metric introduced by Huang, Siegel and Yuan (HSY) [11] could be checked inour calculations. Finally, the so-called bi-adjoint scalar amplitude of CHY formula could besimply produced by calculations of current algebras of closed strings. The gauge dependence of chiral string theory here is of great importance. We first introduce S and T for short: S = { Q, ( b − ¯ b ) } = X ′ · P T = { Q, ( b + ¯ b ) } = 12 ( P + 1 α ′ X ′ ) (1)In the usual string field theory, δ ( f ) / { Q, f } and δ ( f ) δ { Q, f } could be expressed as (see also[14, 15]): δ ( f ) { Q, f } = Z ∞ dτ d ˜ c e ˜ cf − τ { Q,f } δ ( f ) δ ( { Q, f } ) = I π dσ πi d ˜ c e ˜ cf − σ { Q,f } (2)Identify f with either b + ¯ b or b − ¯ b and use δ ( b ± ¯ b ) = b ± ¯ b . Thus, like the usual ”plumbing”analogy in string field theory to get CFT (or, if you like, think of changing from interac-tion picture to Heisenberg picture), we have N unintegrated vertex operators with c ghostinsertions in vertex V s, U = ( c + ¯ c )( c − ¯ c ) V (3)Here, we consider V ∼ e ik · X in this section. The propagator with respect to S and T is alsoeasy to calculate: ∆ = b + ¯ b T ( b − ¯ b ) δ ( S ) (4)Thus, one would get the integrated vertex operators by sandwiching two propagators tocancel the c ghosts and leave the integrations over the corresponding space, in this case( σ, τ ) space. To change into chiral string boundary conditions in conformal gauge, one would effectivelyget the sign-change of the b ghosts, namely,¯ b → − ¯ b S and T . Therefore, the propa-gator changes into ∆ c = b − ¯ b S ( b + ¯ b ) δ ( T ) (5)while the unintegrated vertex operators U are the same. Here, one still has the fact that thepropagators acting on unintegrated vertex operators gives integrated vertex operators. HSZ gauge is a singular gauge choice which is not conformal as the gauge choice discussedabove. We first take the conformal gauge of chiral strings with the same propagators ∆ c andthen use the HSZ gauge before integration. As shown in details in later discussions of thispaper, the HSZ gauge is to modify the propagator h XX i up to an appropriate regulator bysimply transforming the coordinates: z → z ¯ z → ¯ z − βz (6)The CHY δ -function would appear after integration over ¯ z .Noted that we still have S and T playing the roles as propagators. The additionalsingular gauge choices provide the localization of positions of each vertex operator and theintegrations over ¯ z express the localization in terms of δ -functions. By choosing the same gauge of S (Siegel gauge) as usual but different gauge choice of T ,MS formalism has the propagator only depending on S .∆ MS = b − ¯ b S (7)Since the unintegrated vertex operators are sandwiched by propagators ∆ MS to get integratedvertex operators, one only obtains the σ integrations (after relabeling, σ is changed to be z )and leaves the T separated into vertex operators insertions, which is effectively the inversepicture-changing operator: Υ = ( b + ¯ b ) δ ( T ) (8)The propagator ∆ MS and inverse picture-changing operator Υ together cancel the c ghostsof the unintegrated vertex operators ( U O and ¯ U O correspond to open string): U = ( c + ¯ c )( c − ¯ c ) e ik · X = ( c + ¯ c )( c − ¯ c ) V = U O ¯ U O . Before sandwiching the propagators ∆ MS on both sides, the inverse picture-changing oper-ator collides on the unintegrated vertex operator first: W = lim ǫ → Υ( ǫ ) U (0) (9)4sing S as the regulator to regularize the separation of ǫ , one gains lim ǫ → δ ( T ( ǫ )) e ik · X (0) = lim ǫ → Z dτ e τ T ( ǫ ) e ik · X (0) = lim ǫ → Z dτ : e τ T ( ǫ )+ ik · X (0)+ τk · Pǫ := lim ǫ → ( δ ( k · P ( ǫ )) ǫ + O ( ǫ )) e ik · X (0) Then, use ( b + ¯ b )( σ )( c + ¯ c )( σ ) ∼ σ − σ = 1 ǫ (10)to get W = lim ǫ → Υ( ǫ ) U (0) = lim ǫ → ǫ ( δ ( k · P ( ǫ )) ǫ + O ( ǫ ))( c − ¯ c ) e ik · X (0) ∼ ( c − ¯ c ) δ ( k · P (0)) e ik · X (0) (11)Then, sandwiching those vertex operators by the propagators,∆ MS W ∆ MS W ∆ MS W ∆ MS . . . , (12)would lead to the MS formalism:(a) ( b − ¯ b ) cancels ( c − ¯ c ) in W ; namely, for the ghost part {W , b − ¯ b } = V ⇒ ( b − ¯ b ) W ( b − ¯ b ) = V ( b − ¯ b ) (13)(b) S provides the z -integrations at each point z of integrated vertex operator after relabeling σ to be z .Noted that there is no need of α ′ limit in this calculation. When calculate the MS prescription in pure spinor formalism, the b ghost in pure spinorformalism is composite (there is no c ghost in pure spinor formalism).For open string, considering { Q, b ( z ) U ( z ) } = T ( z ) U ( z ) ∼ z − z ∂U ( z )= 1 z − z [ Q, V ( z )] , Since S provides all the propagations, X ( σ ) · P ( σ ) ∼ σ − σ , b ( σ ) c ( σ ) ∼ σ − σ and ¯ b ( σ )¯ c ( σ ) ∼ σ − σ are used and σ is relabeled as z in this subsection.
5e already used [
Q, V ] = ∂U in the last line of calculation. One gets (the BRST trivial partis absorbed into the commutator with Q ) b ( z ) U ( z ) ≈ z − z V ( z ) + [ Q, . . . ] (14)or more general ( b ( z ) + w ( z ))( Q + U ( z )) ≈ T ( z ) + 1 z − z V ( z ) . (15)The BRST trivial term in (14) is due to the background contribution to b in (15); furthermore, w , U and V are the background contributions to b , Q and T , respectively.Noticed that the former calculations also imply { U, b } = V and further b U b = V b . (16)For closed strings, vertex operators and ghosts could be separated into two parts, whichgives: ( b + ¯ b ) U ¯ U ∼ z ( V ¯ U − U ¯ V ) (17) { b − ¯ b , V ¯ U − U ¯ V } ∼ V ¯ V (18)for vertex insertions (like (10) in Bosonic case) and propagators (analogous to (13)), re-spectively. And the discussion of δ ( T ) insertion of the integrated vertex operators is thesame. Pure spinor formalism is based on a worldsheet conformal field theory (CFT) with fields X m , θ α and ghost λ α with the corresponding conjugate momenta, m = 0 , . . . , α =1 , . . . ,
16 as in the usual pure spinor formalism. Then, the worldsheet action in a flat back-ground is given by [8, 16, 17]: S = Z d z ( 12 ∂X m ¯ ∂X m + p α ¯ ∂θ α + ¯ b∂ ¯ c ) + S λ + S J , (19)where S λ and S J are the actions for λ α and J I . And the ghost is constraint further by λγ m λ = 0 . (20)6ith the definition d α = p α −
12 ( γ m θ ) α ∂X m −
18 ( γ m θ ) α ( θγ m ∂θ ) (21)Π m = ∂X m + 12 ( θγ m ∂θ ) . (22)one can easily get the OPEs of those fields: X m ( y ) X n ( z ) ∼ − α ′ η mn ln | y − z | , p α ( y ) θ β ( z ) ∼ α ′ δ βα y − z (23) d α ( y ) d β ( z ) ∼ − α ′ y − z γ mαβ Π m , d α ( y )Π m ( z ) ∼ α ′ y − z ( γ m ∂θ ) α (24) d α ( y ) ∂θ β ( z ) ∼ α ′ ( y − z ) δ βα , Π m ( y )Π n ( z ) ∼ − α ′ ( y − z ) η mn . (25)and the OPEs of current algebra correlated to S J : J I ( y ) J K ( z ) ∼ kδ IK ( y − z ) + f IKL J L ( z )( y − z ) (26)here f IKL is the gauge group structure constant. For any superfield F ( X m ( z ) , θ α ( z )), theOPEs satisfy as follows: Π m ( y ) F ( z ) ∼ − α ′ y − z ∂ m F ( z ) (27) d α ( y ) F ( z ) ∼ α ′ y − z D α F ( z ) (28)with D α := ∂∂θ α + ( γ m θ ) α ∂ m the superderivative. And recall: A α = 12 ( θγ m ) α ǫ m + 13 ( θγ m ) α ( θγ m ξ ) + 116 ( θγ m ) α ( θγ mpq θ ) ∂ q ǫ p + . . .A m = ǫ m + ( θγ m ξ ) + 14 ( θγ pqm θ ) ∂ q ǫ p + 112 ( θγ qpm θ )( θγ q ∂ p ξ ) + . . .W α = 110 γ αβm ( D β A m − ∂ m A β ) F mn = ∂ [ m A n ] (29)Here, ǫ m is the gluon polarization vector and ξ α is the wavefunction for gluino. Thus, thevertex operators of open strings, closed strings and heterotic strings could be easily writtenas:Open: U O = e ik · X ( z ) λ α A α ( θ ) V O = e ik · X ( z ) ( ∂θ α A α + Π m A m + d α W α + 12 N mn F mn ) (30)7losed: U C := e ik · X ( z, ¯ z ) λ α A α ( θ ) λ ¯ β A ¯ β (¯ θ ) V C := e ik · X ( z, ¯ z ) ( ∂θ α A α + Π m A m + d α W α + 12 N mn F mn ) ⊗ ( ¯ ∂θ ¯ β A ¯ β + ¯Π m ¯ A m + d ¯ β W ¯ β + 12 ¯ N mn ¯ F mn ) . (31)Heterotic: U H := e ik · X ( z, ¯ z ) λ α A αI ( X, θ )¯ c ¯ J I V H : = e ik · X ( z, ¯ z ) ( ∂θ α A αI + Π m A mI + d α W αI + 12 N mn F mnI ) ¯ J I (32)The heterotic string vertex operators (32) are only for the super-Yang-Mills amplitude butnot for supergravity amplitude. All U s stand for the unintegrated vertex operator and all V sare for integrated vertex operators. Here, I indices are group index. Thus, the expressionsof four point amplitude of closed (heterotic) strings are: A = h U C ( H )1 ( z ) U ( C ( H )2) ( z ) U C ( H )3 ( z ) Z d z V C ( H )4 ( z ) i . (33)We also need the useful notations in the following calculations: A KN ( z ij ) = h e ik · X ( z , ¯ z ) e ik · X ( z ) , ¯ z e ik · X ( z , ¯ z ) e ik · X ( z , ¯ z ) i = Y i Before integrating over ¯ z , we need to take the singular HSZ gauge limit. As shown in [9], wetake the HSZ gauge instead of the usual conformal gauge: z → z ¯ z → ¯ z − βz. (46)Then the modification of conformal gauge propagator h XX i leads to the substitution:1 z → z z → βz + 1 β ¯ zz . (47)When β → ∞ , we only keep the highest order of β , namely,1¯ z → βz . (48)Note that we would have a minus in front (up to a regulator), if we naively take the HSZgauge limit (46): 1¯ z → − ( 1 βz + 1 β ¯ zz ) (49)but this minus sign is already absorbed into the “sign-flip” caused by the change of boundaryconditions as shown by HSY [11] (see also (62)).First, change the boundary condition, which receives a minus sign in front of the logarithmof ¯ z part, ln | z ij | = 12 (ln z ij + ln ¯ z ij ) → 12 (ln z ij − ln ¯ z ij )= 12 ln z ij ¯ z ij , (50)and substitute the HSZ gauge into the Koba-Nielsen factor,ln z ij ¯ z ij → ln(1 + ¯ z ij βz ij ) ∼ ¯ z ij β z ij , (51)10hen, we obtain the reduced form of Koba-Nielsen factor: A KN = Y i In this paper, we showed the BRST equivalence between MS prescription [19] and ours,and we obtain the amplitude without explicit α ′ dependence rather than α ′ → ∞ or α ′ → δ functions in vertex operator we introduce the formulawith the dependence on both holomorphic and anti-holomorphic part just like the standardcalculations of closed and heterotic strings in conformal gauge. Then, after taking thesingular gauge limit, the integration of Koba-Nielsen factors give us the constraints in δ -functions which coincides with the CHY scattering equations; meanwhile, by changing theboundary conditions, the rest of the anti-holomorphic part turns into the holomorphic parttimes polynomials of singular gauge parameters, which cancels the gauge dependence inside δ -functions.Our method could be easily generalized to N -point scattering amplitude with g loops by:(a) Calculate the amplitude in usual conformal gauge;(b) Take some singular gauge limit and then integrate over ¯ z , which will only give us the δ -functions;(c) The integral over z will lead to substitution of the solutions with respect to the constraintsinside δ -functions (scattering equations) into the integrand.We could also propose that those singular gauge limits are related to some special quasi-conformal symmetry by Beltrami derivatives. 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