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Supersymmetry Breaking Effects using the Pure Spinor Formalism of the Superstring

The SO(32) heterotic superstring on a Calabi-Yau manifold can spontaneously break supersymmetry at one-loop order even when it is unbroken at tree-level. It is known that calculating the supersymmetry-breaking effects in this model gives a relatively accessible test case of the subtleties of superstring perturbation theory in the RNS formalism. In the present paper, we calculate the relevant amplitudes in the pure spinor approach to superstring perturbation theory, and show that the regulator used in computing loop amplitudes in the pure spinor formalism leads to subtleties somewhat analogous to the more familiar subtleties of the RNS approach.


Introduction
Like the Green-Schwarz (GS) formalism, the pure spinor formalism for the superstring [1] has the advantage over the Ramond-Neveu-Schwarz (RNS) formalism that the worldsheet action is manifestly spacetime supersymmetric. So there is no sum over spin structures in loop amplitudes and Ramond-Ramond backgrounds can be conveniently described. However, in contrast to the GS formalism, the worldsheet action in a flat background is quadratic in the pure spinor formalism and κ-symmetry is replaced by a BRST symmetry, generated by a charge Q whose square vanishes. So covariant quantization is straightforward and some multiloop amplitudes can be explicitly computed.
In a general loop computation, a potential subtlety arises. In the pure spinor formalism, if one attempts to compute loop amplitudes using the manifestly supersymmetric action that can be used to determine the spectrum, one has difficulty interpreting the necessary integration over zero-modes. To make sense of the zero-mode integrations, one needs to add to the action a BRST-trivial regulator term {Q, . . .}. The regulator term is not spacetime supersymmetric but since it is BRST-trivial, its supersymmetry variation is also BRST-trivial. So formally one expects loop amplitudes to be spacetime supersymmetric. However, since a BRST-trivial term leads to an exact form on moduli space, there is now an integration by parts on moduli space required to prove spacetime supersymmetry.
There is a possibility that the proof might fail in some cases because surface terms will appear in the integration by parts. When the proof does work, this might depend on a delicate treatment of the region at infinity in moduli space.
In the RNS approach to superstring perturbation theory, spacetime supersymmetry is not manifest and its proof always involves an integration by parts on supermoduli space. A delicate treatment is needed and requires working with super Riemann surfaces. In many superstring compactifications with spacetime supersymmetry, the subtleties do not appear to arise in practical computations of low order. A simple exception is the SO(32) heterotic string compactified on a Calabi-Yau manifold. Supersymmetry is unbroken at tree level, but generically it is spontaneously broken at the one-loop order [2]. It turns out that the supersymmetry-breaking effects that arise at 1-and 2-loop order 3 give interesting test cases for superstring perturbation theory in the RNS formalism: they are simple enough that one can analyze them, but subtle enough so that it is hard to do so correctly (and get the right answers) in an approach based on trying to reduce everything to ordinary rather than super Riemann surfaces. Some of this was explained in the early literature [3][4] [5][6] [7], and a more comprehensive analysis has been made recently [8].
The purpose of the present paper is to analyze these same amplitudes in the pure spinor approach to superstring perturbation theory. We will find that the regulator of the pure spinor approach leads to subtleties somewhat analogous to the subtleties of super Riemann surfaces in the RNS formalism. This result is not unexpected since otherwise it would not be possible for a model that is supersymmetric at tree level to show spontaneous supersymmetry breaking at the 1-loop level.
In section 2 of this paper, we will review the pure spinor description of the heterotic superstring on a Calabi-Yau manifold with spin connection embedded in the gauge group.
In a background with anomalous U(1) gauge field, the auxiliary scalar D in the U (1) vector multiplet gets a one-loop expectation value D = 0 which spontaneously breaks supersymmetry.
In section 3, we compute in this background the one-loop mass splitting of bosons and fermions of charged multiplets. To regularize the one-loop two-point amplitude, one needs to put a cutoff so that the locations of the two vertex operators do not coincide. And BRST invariance in the presence of this cutoff requires the inclusion of a contact term.
This contact term gives a contribution to the mass of charged bosons which is proportional to D , but does not affect the mass of charged fermions.
In section 4, we compute in this background the two-loop cosmological constant (or vaccum energy). To regularize the zero-point two-loop amplitude, one needs to introduce a cutoff in the limit that the genus-two surface degenerates into two genus-one surfaces.
BRST invariance again requires a contact term, and this contact term gives a contribution to the cosmological constant which is proportional to D 2 .
To some extent, our computations in this paper are experimental, since the subtleties of the pure spinor regulator have not been fully explored. Perhaps the examples we study will help in doing so.
(2.1) (The nonminimal variables are not needed to construct a manifestly supersymmetric model that can be quantized to give the superstring spectrum, but they facilitate the construction of the regulator.) The right-moving worldsheet variables are the same as in the RNS formalism and include the fermionic spin-half variables ξ r for r = 1 to 32 and the Virasoro The pure spinor formalism can be defined in any consistent supergravity background and the worldsheet action is where Z M = (x m , θ α ) are the N = 1 d = 10 superspace variables, 1 2 (λγ ab w) are the SO(9,1) Lorentz currents for the left-moving pure spinors and a, b = 0 to 9 are tangentspace vector indices, J i = ξ r T i rs ξ s for i = 1 to 496 are the SO(32) currents for the right-moving fermions, T i rs are the SO(32) adjoint matrices, (G M N , B M N , E α M , Ω ab M , Φ) are supergravity superfields for the metric, antisymmetric tensor, spinor vierbein, spin connection and dilaton, (A M i , W α i , F ab i ) are the super-Yang-Mills superfields for the gauge field, spinor field strength and vector field strength, and r is the worldsheet curvature scalar.
Although it will be unnecessary in this paper, the nonminimal variables in w α ∂λ α + s α ∂r α can be made covariant with respect to local Lorentz transformations by coupling them to the spin connection in a BRST-invariant manner as described in [9] [10].
where the second term in (2.3) ensures that the cohomology is independent of the nonminimal variables. The BRST transformations of the superspace variables Z M = (x m , θ α ) are is the spinor component of the inverse supervierbein, so the BRST transformation of a scalar spacetime superfield Φ(Z) is ∂Z M are the spacetime supersymmetric derivatives in the curved background.
For compactification on a Calabi-Yau manifold, the ten x m variables split into (x Q , y J , y J ) for Q = 0 to 3 and J, J = 1 to 3, and the 32 right-moving ξ r variables split into (ξ J , ξ J , ξ T ) for T = 1 to 26. It will be convenient to keep the 16-component SO (9,1) spinor index α for the spacetime spinors. The worldsheet action in this background is where ∇ = ∂ + 1 4 (∂y J Ω ab J (y)γ ab + ∂y J Ω ab J (y)γ ab ), (2.5)

6)
F JKi are auxiliary fields which will be relevant for supersymmetry breaking, and . . . denotes terms which are higher-order in θ α . The higher-order terms in θ α will not be needed here and can be determined by requiring BRST invariance of the worldsheet action, or alternatively by solving the superspace equations of motion for the supergravity and super-Yang-Mills superfields in the Calabi-Yau background.

Vertex operators
In the d = 10 heterotic string, the supergravity and super-Yang-Mills onshell states are described by the BRST-invariant vertex operators of dimension (0,1) and ghost-number (1,0): and (a mi (x), ζ α i (x)) are supergravity and super-Yang-Mills fields, and . . . are terms higher-order in θ α which are related by BRST invariance to the lowerorder terms. For example, in a flat background, the BRST transformation of the term The integrated vertex operator for this auxiliary field appears through the covariant derivative (2.6) in the heterotic sigma model action of (2.4). And the unintegrated vertex operator for this auxiliary field in the pure spinor formalism can be obtained from the term proportional to (2.11) in (2.10) which where . . . denotes terms higher-order in θ α . This can be compared with the vertex operator for D in the RNS formalism, which is [3] (2.14) To show that V D of (2.13) correctly describes the auxiliary field, note that the BRST transformations of (2.7) imply to be the string field and computing the quadratic term in the tree-level string field theory action 1 2g 2 s ΦQΦ using the measure factor one obtains the appropriate tree-level term in the field theory action The action of (2.17) can also be obtained by starting with the string field theory action in a flat d=10 background and then setting D = g JJ F JJ and restricting the dependence on the six Calabi-Yau directions. In a flat background, the quadratic term in the string field theory action is which gives the usual d=10 Maxwell action after solving the equation of motion for the auxiliary field F mn and plugging back into the action.
At one loop, it will be shown in subsection 3.3 that the auxiliary field D couples linearly to the Calabi-Yau background field-strengths F JK in the form (2.20) Combining the tree-level term of (2.17) with the one-loop term of (2.19), the value of D is shifted from the classical answer g JJ F JJ (which vanishes in Calabi-Yau compactification at the classical level) and one learns that D has the expectation value which spontaneously breaks spacetime supersymmetry when p of (2.20) is nonzero. In the next two sections, we will show how to compute the effects of this supersymmetry breaking using the pure spinor formalism.

One-Loop Mass Splitting
In this section, the two-point one-loop amplitude of charged bosons will be shown to be nonzero after including a contact term required for BRST invariance. The analogous twopoint one-loop amplitude of charged fermions related by supersymmetry is zero, implying that spacetime supersymmetry has been broken.

BRST-invariant prescription
The two-point one-loop amplitude is naively computed in the pure spinor formalism by where V (1) is the unintegrated dimension (0,1) vertex operator of the first external state, is the insertion of the composite b ghost contracted with the Beltrami differential µ dual to the Teichmuller parameter τ , and N is a regulator defined as N = e QΛ for some Λ.
The regulator N = e QΛ is needed to perform the functional integral over the noncompact zero modes of the pure spinors (λ α , λ α ) and their conjugate momenta (w α , w α ), and the precise form of Λ is irrelevant as long as the functional integral is regularized and BRSTtrivial states decouple. For example, choosing Λ = −ρ(λ α θ α + w α s α ) where ρ is a positive constant implies that which regularizes the functional integral over the (λ α , λ α , w α , w α ) zero modes for any choice of ρ.
Because U (2) (z 2 ) may have a singularity when z 2 → z 1 , one needs to introduce a cutoff ǫ in the integral over ǫ d 2 z 2 so that z 2 is integrated over the entire genus-one surface except for a small disk of radius ǫ around z 1 . Similarly, since the loop amplitude may be singular when the torus becomes infinitely thin, one needs to put a cutoff ǫ ′ in the integration over the Teichmuller parameter. In the presence of these cutoffs, BRST invariance is not manifest since QU (2) = ∂V (2) and Qb = T . So changing Λ in N will produce the term The second term in (3.4) gives a surface term at the boundary of moduli space where the torus degenerates into an infinitely thin cylinder. From a field theory point of view, one would say that the momentum flowing through this channel is an integration variable and thus is generically off-shell. In general, in string theory, there is never a surface term arising from a degeneration at which the momentum is generically off-shell. The clearest explanation of this involves incorporating the Feynman iε in string theory (see [11] on this point, and see [12] for an earlier discussion of the Feynman iε in string theory).
The first term in (3.4) gives where C is a contour of radius ǫ around z 2 = z 1 and in the BRST cohomology at ghost-number (2, 0). In the heterotic superstring, the BRST cohomology at ghost-number (1, 0) describes physical vertex operators such as (2.8), and the BRST cohomology at ghost-number (2, 0) describes the duals to these physical vertex operators. If one assumes that δΛ is a spacetime scalar, V (3) must also be a spacetime scalar in order to contribute to (3.5). But the only spacetime scalar in the BRST cohomology at ghost-number (2, 0) is the dual vertex operator for the dilaton, i.e.
Note that the dual vertex operators in the super-Yang-Mills multiplet are either spacetime spinors or vectors. Since we are assuming the external states are super-Yang-Mills states which do not carry ∂x m dependence, the vertex operator of (3.7) cannot be present in Ω.
So we will have Ω = QΣ for some Σ (in our examples, we will see explicitly what Σ is) and where the contribution from Qb in (3.8) can be ignored for the same reason that the second term in (3.4) vanishes.
So although the amplitude prescription of (3.1) is not manifestly BRST-invariant, it can be modified to the BRST-invariant amplitude prescription where Σ is defined by (3.10)

Mass splitting
where . . . denotes higher-order terms in θ α . Using the OPE ξ T (z 1 )ξ T ′ (z 2 ) → (z 1 − z 2 ) −1 δ T T ′ , one finds that Using the BRST-invariant prescription of (3.9) for the two-point amplitude involving charged bosons, the first term is proportional to k 2 and therefore vanishes when the states are onshell. Indeed, in d = 10, the charged boson is a gauge field A J which can only appear in the gauge-invariant After dimensional reduction, this implies that the two-point amplitude must be proportional to η mn g JJ F mJ F nJ = k 2 g JJ A J A J . However, the second term of (3.9), is nonvanishing, as we will argue in the next section.
For charged fermions in the same supersymmetric multiplet as the bosons, Lorentz invariance and charge conservation imply that the two-point amplitude must be proportional to γ αβ Q k Q . But γ αβ Q k Q vanishes when the states are onshell, so the two-point amplitude for charged fermions is zero and there is a one-loop mass splitting if the one-loop mass shift of the bosons is non-zero.

One-loop expectation value
As explained in [13], the one-loop open superstring amplitude in the pure spinor formalism is computed by where b is the composite operator satisfying {Q, b} = T and Using the zero mode analysis of [13], it is easy to verify that the contribution to V D to lowest-order in α ′ comes if three factors of the U JK vertex operator contribute from the worldsheet action. So instead of computing the one-point amplitude V D in a Calabi-Yau background, one can instead compute the four-point amplitude in a flat background. As in the four gluon one-loop amplitude computation in [13], the only contribution to (3.18) comes if V D contributes three θ α zero modes, each U JK contributes one θ α zero mode and one p α zero mode, the b ghost contributes one r α zero mode and two p α zero modes, and N contributes 10 θ α zero modes, 10 r α zero modes, 11 s α zero modes, and 11 p α zero modes.
To compute the index contractions of (3.18), it is useful to note that V D of (2.13) resembles the term of order (θ) 3 in the vertex operator of an on-shell gluon with constant field-strength in the direction F JJ = g JJ . In ten uncompactified dimensions, the massless string field at ghost-number one has the form where V only depends on the worldsheet zero modes. QV = 0 and δV = QΩ implies that A α (x, θ) is the on-shell super-Yang-Mills spinor gauge field which can be gauge-fixed to the form where ... includes components which are related to derivatives of the on-shell gluon a m (x) and gluino ξ β (x) and F mn = ∂ m a n − ∂ n a m . In the RNS formalism, one can perform a similar analysis by working in light-cone gauge and twisting the eight light-cone ψ j variables to have integer spin (as in light-cone Green-Schwarz) so that they become periodic variables on the worldsheet. In this case, one needs to absorb eight ψ j zero modes, so the RNS vertex operator of (2.14) can have a non-zero one-loop amplitude only if the Calabi-Yau background field-strengths contribute three factors of dzF JK ψ J ψ K . As in the pure spinor computation, the V D computation is related to the four gluon computation since zero mode analysis implies that one can ignore the x-dependent terms in (3.23) and replace the unintegrated gluon vertex operator in the four gluon one-loop computation with (2.14).
In [3], the result V D = p was argued to be exact to all orders in α ′ by using the lightcone Green-Schwarz formalism to show that massive string states do not contribute to V D .
In the covariant RNS formalism, a similar argument can be made after choosing a special gauge for the worldsheet supermoduli such that the covariant RNS computation reduces to the light-cone computation. It would be interesting to look for a similar argument using the pure spinor formalism, perhaps involving a special choice for the regulator N . The computation that we have presented here is only valid to lowest order in α ′ .

Simpler example of mass splitting
As discussed in [8], the one-loop two-point function of a dilatino and a gaugino also exhibits a supersymmetry-breaking mass shift. In the RNS formalism, this supersymmetrybreaking effect is more straightforward than the example treated above: many of the usual subtleties of superstring perturbation theory do not arise, because the relevant super moduli space has only one odd modulus. As we will now show, this example is also more straightforward in the pure spinor approach, in this case because it can be computed without introducing any cutoffs.
The pure spinor prescription for the one-loop dilatino-gaugino two-point amplitude is where V (1)α = γ αδ n (λγ m θ)(γ m θ) δ ∂x n is the unintegrated vertex operator for the dilatino at zero momentum, and U is the integrated vertex operator for the gaugino at zero momentum.
After contracting the ∂x n in V (1)α with the ∂x m in U (2) β and integrating over d 2 z 2 , one obtains which is proportional to where V D is defined in (2.13) and p is defined in (2.20).

SUSY goldstino
Here we will explore what goes wrong with a direct attempt to prove space-time supersymmetry. (This discussion will be a pure spinor analog of the rather intricate RNS discussion given in section 4 of [8].) The naive computation to show that the one-loop mass shift is the same for supersymmetric partners involves looking at A = (γ 0123 ) β α A α β where α and β are spinor indices which are SU(3) singlets and where cV (1) is the unintegrated vertex operator for the boson, U (2)α is the integrated vertex operator for the fermion, and q β is the susy charge whose contour is integrated around the two vertex operators. The right hand side of (3.27) is the sum of two contributions in which q β is commuted with one vertex operator or the other. The sum of these two contributions is the difference of boson and fermion mass shifts, so A = 0 is the condition for unbroken supersymmetry of these amplitudes. To try to prove that A = 0, we pull the contour of q β off of the vertex operators. The only contribution comes from the commutator of q β with the regulator N of (3.2), which is [q β , N ] = −ρ r β N = Q(ρ λ β N ).
So if (V (1) (z 1 ) d 2 z 2 U (2)α (z 2 )) were BRST-closed, the BRST operator Q could be pulled off of (λ β N ) and A α β would vanish. However, as discussed in subsection 3.1, (V (1) (z 1 ) d 2 z 2 U (2)α (z 2 )) is not BRST-closed and satisfies Q( (3.28) If Ω α were equal to QΣ α for some Σ α , one could define a regularized version of (3.27) as in (3.9) to be  , Ω α is not BRST-trivial and one cannot write Ω α = QΣ α . Defining Ω α = QΣ α + hV (3)α for some constant h, one finds that Performing the functional integral over the pure spinors using the regulator of (3.2), one obtains that A α β is proportional to h(γ 0123 ) α β . So the difference of the mass-shift computed by A = (γ 0123 ) β α A α β is proportional to h, which measures the coupling of the bosonic and fermionic states described by V (1) and V (2)α with the SUSY goldstino whose dual vertex operator V (3)α is defined in (3.30).

Two-Loop Cosmological Constant
In the RNS framework, a natural procedure to evaluate the two-loop dilaton tadpole or cosmological constant is to integrate over the odd moduli keeping fixed the super period matrix [14]. This procedure actually gives what one might call the bulk contribution to the cosmological constant. In general, as explained in section 3 of [8], one also requires a boundary contribution. 4 In the case of a supersymmetric compactification to four dimensions, this boundary contribution is a universal multiple of V D 2 , where V D is computed in genus 1. It was conjectured in [8] that in an arbitrary supersymmetric compactification to four (or more) dimensions, the bulk contribution to the genus 2 cosmological constant vanishes and the full answer comes from the boundary contribution. It is actually rather tricky to prove this in the RNS framework. The conjecture has been supported by explicit calculations in some orbifold compactifications to four dimensions with N = 1 supersymmetry [15], and a general proof (based on an analysis of the super period matrix with Ramond punctures) will appear elsewhere.
We will aim here to explore the analogous issues in the pure spinor formalism. In this, we have to remember that in subsection 3.3, we determined the one-loop expectation value V D in the pure spinor formalism only to the lowest non-trivial order in α ′ . Similarly, at a certain point below, we will have to make an argument that is valid only to lowest non-trivial order in α ′ .
In the pure spinor formalism, the naive formula for the two-loop cosmological constant where N = e QΛ is the regulator needed to perform the functional integral over the noncompact pure spinor zero modes and µ j are the Beltrami differentials dual to the three complex Teichmuller parameters τ j of the genus-two surface.
As in the one-loop amplitude of the previous section, the possible presence of singularities implies that one needs to introduce cutoffs near the boundary of the moduli space of the genus-two surface. The genus-two surface can degenerate in two different ways: either by one of the two handles becoming infinitely thin or by the surface separating into two genus-one surfaces connected by an infinitely thin tube. The first type of degeneration is harmless for the same reason as in the one-loop amplitude; the momentum flowing through the degenerating handle is an integration variable. However, the second type of degeneration needs to be carefully treated and will give a contribution when spacetime supersymmetry is spontaneously broken at one-loop.
We parametrize the genus 2 moduli space with parameters τ 1 , τ 2 , τ 3 such that the separating degeneration occurs for Im τ 3 → ∞ and that in this limit, transforms as where the genus-two surface has been factorized into two genus-one surfaces connected by a thin tube and I is a sum over all possible states going through the tube.
It will be assumed that for Im τ 3 → ∞, the original regulator N factorizes as the product of two regulators N 1 N 2 where N 1 = e QΛ 1 is inserted on the first genus-one surface at y 1 , N 2 = e QΛ 2 is inserted on the second genus-one surface at y 2 , and Λ 1 has been shifted by δΛ 1 . This assumption is necessary since when the surface degenerates, the two holomorphic zero modes of the conformal weight one variables (w α , d α , s α ) separate onto the two different surfaces. So the regulator of (3.2) only provides the correct zero mode dependence if it factorizes into two regulators on the different surfaces. This is analogous to the necessity in the RNS formalism of distributing the picture-changing operators on the different surfaces.
In eqn. (4.2), the state going into the thin tube from one side is represented by the vertex operator cV * I (z 1 ) of ghost-number (2, 1) and the state going out of the thin tube on the other side is represented by the vertex operator cV I (z 2 ) of ghost-number (1, 1). The vertex operators cV * I and cV J are defined to satisfy the dual relation using the tree-level measure factor of (2.16).
As discussed in subsection 2.
where p is defined in (2.20). And since QV D of (2.15) satisfies V D (QV D ) = 1 using the measure factor of (2.16), V * D = QV D is consistent with (4.3). So δA of (4.2) is equal to One can therefore define a manifestly BRST-invariant two-loop cosmological constant with the prescription where we have used the lowest order result for the one-loop expectation value V D .
Finally, it will be argued by analyzing zero modes of θ α that the first term on the right hand side of (4.6) vanishes to lowest non-trivial order in α ′ . Under the SO(3, 1) × SU (3) × U (1) decomposition of SO(9, 1), there are four θ α zero modes which are SU (3) singlets and cannot come from the worldsheet action of (2.4). So for the first term to be non-vanishing, these four zero modes must all come from the e −θ α r α term in the regulator N of (3.17). In fact, the argument will only require that at least one θ α zero mode must come from N .
So we have shown that for any function f satisfying f (0) = 0. This is only possible if after performing the functional integral over all worldsheet variables except for λ + and λ + , ( µ j b) = ce −λ + λ + (4.12) for some proportionality constant c.
So we have shown that after including the contact term required for BRST-invariance, the two-loop cosmological constant is equal to g 2 s p 2 , at least to lowest non-trivial order in α ′ . This result is expected to hold to all orders in α ′ and it would be desirable to find a proof using the pure spinor formalism.