Multiloop Amplitudes of Light-cone Gauge NSR String Field Theory in Noncritical Dimensions

Feynman amplitudes of light-cone gauge superstring field theory are ill-defined because of various divergences. In a previous paper, one of the authors showed that taking the worldsheet theory to be the one in a linear dilaton background $\Phi=-iQX^{1}$ with Feynman $i\varepsilon$ $(\varepsilon>0)$ and $Q^{2}>10$ yields finite amplitudes. In this paper, we apply this worldsheet theory to dimensional regularization of the light-cone gauge NSR superstring field theory. We concentrate on the amplitudes for even spin structure with external lines in the (NS,NS) sector. We show that the multiloop amplitudes are indeed regularized in our scheme and that they coincide with the results in the first-quantized formalism through the analytic continuation $Q\to0$.


Introduction
Since the light-cone gauge superstring field theory [1][2][3][4][5][6] takes a simple form, this theory is expected to be very useful in studying superstring theory. In a series of papers [7][8][9][10][11][12][13][14], using the light-cone gauge closed NSR superstring field theory, we have studied the contact term divergences [15][16][17][18][19]. We have pointed out that the contact term divergences can be regularized by shifting the central charge of the light-cone gauge worldsheet theory to a sufficiently large negative value [7]. We refer to this type of regularization as the dimensional regularization, since the central charge is directly related to the spacetime dimensions in string theory.
We have considered mainly two ways to shift the central charge in the regularization: The one is to naively shift the number of the transverse dimensions d − 2; The other is to add a superconformal field theory with central charge large negative to the worldsheet theory.
Recently, one of the authors has proposed another prescription [20], in which the string theory in a linear dilaton background Φ = −iQX 1 is considered so that the central charge of the system becomes 12 − 12Q 2 . In [20], the divergences which appear in the amplitudes of this theory have been thoroughly studied. It has been shown that the Feynman amplitudes of light-cone gauge superstring field theory in the linear dilaton background are indeed finite, when the theory is with the Feynman iε (ε > 0) and Q 2 > 10.
In this paper, we use this theory to dimensionally regularize and calculate the Feynman amplitudes. In this work, we restrict ourselves to the amplitudes for even spin structure with external lines in the (NS,NS) sector for simplicity. In this scheme, we define the amplitudes as analytic functions of Q and perform the analytic continuation Q → 0. We show that the limit Q → 0 is smooth except the divergences coming from the boundaries of the moduli space and the results coincide with those from the first-quantized method. 3 In order to show the coincidence between our results and those in the first-quantization, we recast the amplitudes into a BRST invariant form in terms of the conformal gauge worldsheet theory. For this purpose, together with the superreparametrization ghosts, we introduce the longitudinal variables with nonstandard interactions, which is the supersymmetric X ± CFT constructed in [9] with the identification d − 10 The organization of this paper is as follows. In section 2, we recapitulate the light-cone gauge superstring field theory in the linear dilaton background constructed in [20]. In section 3, we show that the multiloop amplitudes can be expressed in terms of a BRST invariant worldsheet theory in the conformal gauge. For this purpose, we add the supersymmetric 3 In [21][22][23], Sen has constructed covariant string field theories which reproduce the Feynman amplitudes from the first-quantized approach.
X ± CFT and superreparametrization ghosts to the worldsheet theory. We show that the supercurrents in the light-cone gauge, inserted at the interaction points, can be transformed into the picture changing operators (PCO's) in the conformal gauge and the expressions become BRST invariant. In section 4, we carry out the analytic continuation Q → 0 of the Feynman amplitudes. We show that the results from the first-quantized formalism are reproduced. Namely, the results obtained here coincide with those obtained using the Sen-Witten prescription [24][25][26], up to infrared divergence problems. Section 5 is devoted to conclusions and discussions. In appendix A, the definitions of the Arakelov metric and Arakelov Green's functions are presented. In appendix B, some details of the supersymmetric X ± CFT are given. Formulas used in the text are provided in appendices C and D.

Superstring field theory in linear dilaton background
In this section, we review the light-cone gauge superstring field theory in linear dilaton background constructed in [20]. The string field theory is given for Type II superstring theory formulated in the NSR formalism. The heterotic case can be dealt with in a similar way.

Linear dilaton background
In order to regularize various divergences, we consider the superstring theory in a linear dilaton background Φ = −iQX 1 , with a real constant Q. The worldsheet action of X 1 and its fermionic partners ψ 1 ,ψ 1 on a worldsheet with metric ds 2 = 2ĝ zz dzdz becomes S X 1 , ψ 1 ,ψ 1 ;ĝ zz = 1 8π dz ∧ dz ĝ ĝ ab ∂ a X 1 ∂ b X 1 − 2iQRX 1 and the energy-momentum tensor and the supercurrent, which generate the superconformal transformations, are given as In order to construct string field theory and calculate amplitudes we need the correlation functions of the linear dilaton conformal field theory. Since the fermionic part is just a free theory, we concentrate on the bosonic part. Defining the correlation function of e iprX 1 (Z r ,Z r ) (r = 1, · · · , N) can be calculated on a Riemann surface [20]. Using the Arakelov metric ds 2 = 2g A zz dzdz [27] defined on the surface, it is given as where Z X g A zz denotes the partition function for a free boson on the worldsheet with the Arakelov metric (C.22), and of the linear dilaton conformal field theory.
e ipX 1 thus defined turns out to be a primary field with conformal dimension Notice thatX 1 satisfies if there are no source terms, and thus i∂X 1 (z), i∂X 1 (z) can be expanded as where α 1 n andᾱ 1 n satisfy the canonical commutation relations. The states in the CFT are given as linear combinations of the Fock space states where |p = e ipX 1 (0)|0 . The states and the oscillators satisfy where p|, (α 1 n ) * , (ᾱ 1 n ) * are the BPZ conjugates of |p , α 1 n ,ᾱ 1 n respectively. On the sphere, the correlation function is given by using the worldsheet metric ds 2 = dzdz on the complex plane as Using these, it is straightforward to construct the light-cone gauge superstring field theory action in the background.

Light-cone gauge superstring field theory in linear dilaton background
Let us construct the light-cone gauge superstring field theory based on the worldsheet theory with the variables where the action for X 1 , ψ 1 ,ψ 1 is taken to be (2.1) and that for other variables is the free one. The worldsheet theory of the transverse variables turns out to be a superconformal field theory with central charge The string field is taken to be an element of the Hilbert space of the transverse variables on the worldsheet and a function of (2.14) |Φ(t, α) should be GSO even and satisfy the level-matching condition where L 0 ,L 0 are the zero modes of the Virasoro generators of the worldsheet theory.
The action of the string field theory is given by [7,11] (2.16) The first and the second terms are the kinetic terms with the Feynman iε and Φ(−α)| denotes the BPZ conjugate of |Φ(−α) . The third and the fourth terms are the three string vertices and g s is the string coupling constant. B and F denote the sums over bosonic and fermionic string fields respectively. By the state-operator correspondence of the worldsheet conformal field theory, there exists a local operator O Φ (w,w) corresponding to any state |Φ . on the worldsheet. h r (w r ) gives a map from a unit disk |w r | < 1 to the cylinder corresponding to the r-th external line so that   A Wick rotated g-loop N-string diagram is conformally equivalent to an N punctured genus g Riemann surface Σ. A g-loop N-string amplitude is given as an integral over the moduli space of Σ as [28,29] A (g) [dα] denotes the integration over the moduli parameters and C is the combinatorial factor. In each channel, the integration measure is given as Here T a 's are heights of the cylinders corresponding to internal lines, α A 's denote the circumferences of the cylinders corresponding to the + components of the loop momenta and α I 's and θ I 's are the string-lengths and the twist angles for the internal propagators.
N is given as a path integral over the transverse variables X i , ψ i ,ψ i (i = 1, . . . , 8) on the light-cone diagram. A light-cone diagram consists of cylinders which correspond to propagators of the closed string as mentioned above. On each cylinder, one can introduce a complex coordinate ρ in the same way as the complex coordinate (2.18) is introduced on the three string vertex. The ρ's on the cylinders are smoothly connected except at the interaction points and we get a complex coordinate ρ on the light-cone diagram Σ. The path integral on the light-cone diagram is defined by using the metric (2.19).
ρ is not a good coordinate around the interaction points and the punctures, and the metric (2.19) is not well-defined at these points. F (g) N can be expressed in terms of correlation functions defined with a metric dŝ 2 = 2ĝ zz dzdz which is regular everywhere on the worldsheet, as (2.23) Here S LC X i , ψ i ,ψ i denotes the worldsheet action of the transverse variables and the path integral measure dX i dψ i dψ i ĝzz is defined with the metric dŝ 2 = 2ĝ zz dzdz. Since the integrand should be defined by using the canonical metric ds 2 = ∂ρ∂ρdzdz on the light-cone diagram, we need the anomaly factor e − in the (NS,NS) sector, the light-cone vertex V LC r is given as and z I (r) is defined to be the coordinate of the interaction point at which the r-th external line interacts. The on-shell and the level-matching conditions are It is possible to calculate the right hand side of (2.23). ρ can be given as a function of local coordinate z on Σ as up to an additive constant independent of z. Here E(z, w) is the prime form of the surface, ω is the canonical basis of the holomorphic abelian differentials and Ω is the period matrix. 4 The base point P 0 is arbitrary. There are 2g − 2 + N zeros of ∂ρ and we denote them by . They correspond to the interaction points of the light-cone diagram. Substituting (2.30) into (2.24) yields a divergent result for Γ [σ;ĝ zz ]. We can obtain e −Γ[σ;ĝ zz ] up to a divergent numerical factor by regularizing it as was done in [31].
The divergent factor can be absorbed in a redefinition of g s and the vertex operator. Takinĝ g zz to be the Arakelov metric [27], e −Γ[σ;g A zz ] for higher genus surfaces is calculated in [13] to be up to a numerical constant which can be fixed by imposing the factorization condition. Here For the mathematical background relevant for string perturbation theory, we refer the reader to [30].
The correlation functions of X i , ψ i ,ψ i which appear in (2.23) can be calculated by using the formulas given in [32][33][34][35][36] and (2.4). From the explicit form of the integrand F (g) N , one can see that the amplitude A (g) N suffers from divergences due to the collisions of the interaction points and the degenerations of the surface Σ, if Q = 0. In [20], it was shown that A (g) N becomes finite if one takes Q 2 > 10 and ε > 0. Therefore it is possible to define A (g) N as an analytic function of Q and take the limit Q → 0 as is usually done in dimensional regularization of field theory.

BRST invariant form of the amplitudes
We would like to show that the integrand (2.23) can be rewritten in terms of a correlation function of the conformal gauge variables For Q = 0, the worldsheet theory of the longitudinal variables X ± , ψ ± ,ψ ± becomes socalled X ± CFT, which is defined and analyzed in [9,14]. The results of these references are summarized in appendix B. Using (B.15) and (B.16) there, one can prove Here X ± is the superfield given in (B.3). The supersymmetric contour integral is defined as using the Berezinian integral dθ, and the antiholomorphic version is defined similarly.
−s for the r-th string defined as −s are similarly given for the antiholomorphic sector. The product SSV DDF r is normal ordered as We will further rewrite (3.7) by introducing the ghost variables. The identities satisfied by the ghost correlation functions which should be used here are summarized in appendix C. Taking the metricĝ zz to be the Arakelov metric g A zz and using (3.7), (C.28) and (C.35), the amplitude (2.21) can be rewritten as Here S tot denotes the worldsheet action for the variables X µ , ψ µ ,ψ µ and the ghosts. It is shown in [9] that the worldsheet theory becomes a conformal field theory with vanishing central charge, and we can define the nilpotent BRST operator Q B . The quantities which appear in this expression would be BRST invariant, if e φ T LC F (z I ) and eφT LC F (z I ) were the PCO's. Actually (3.8) can be turned into a BRST invariant form as is the PCO andX (z) is its antiholomorphic counterpart. Here T F denotes the supercurrent for X µ (µ = +, −, 1, . . . , 8). Since the PCO's and the contour integrals of S (z, Z r ) ,S z,Z r do not commute, we need to be a little careful about the definition of the right hand side of (3.9). To be precise, the right hand side of (3.9) should be defined as The contour C r,|ǫ| is a circle with radius |ǫ| around z = z I (r) . One can show and with no operators inside C r,|ǫ| ,C r,|ǫ| , this vanishes. All the other factors are BRST invariant in the usual way and thus one can prove that (3.9) is a BRST invariant expression.
When there exists no r (r = 1, . . . , N) such that z I = z I (r) , we can simply replace the limit (3.14) In the generic situation in which z I , Z r are all distinct, 5 it is not possible for a z I to be equal to z I (r) for more than two r's, except for the tree-level three point amplitudes. It is possible 5 The situation in which some of z I , Z r coincide can be considered as a limit of these generic situations.
to derive the formula as (3.14) in such cases. Thus we can see that the limit ǫ → 0 in (3.11) is not singular and we denote the result by the naive expression (3.9). Eq.(3.9) is proved in appendix D.

The amplitudes from the first-quantized formalism
In recent papers [24][25][26], a way to calculate superstring amplitudes using the PCO's is established. We would like to show that the amplitudes calculated from the light-cone gauge string field theory using the dimensional regularization coincide with those obtained by the method of these papers.

The prescription
In this subsection, we will just briefly explain the prescription given in [26].
In the first-quantized approach using PCO's, an amplitude is expressed by an integral of a correlation function with a fixed number K of PCO insertions over the relevant moduli space M which is assumed to have real dimension n. Each pointṁ ∈ M determines a Riemann surface Σ(m). Let Y be a fiber bundle over M whose fiber is in order to describe the insertions of K PCO's, and X be the subspace of Y which is obtained by omitting the bad points where the spurious singularities arise. A point in X is denoted where the integrand is schematically expressed as [24] ω n (m; (4.1) here denotes the correlation function of the worldsheet theory on Σ(m), V 1 , · · · , V N are BRST invariant vertex operators, m 1 , · · · , m n are the coordinates of M and the subscript n denotes that we should extract the n-form part of this expression.
Unfortunately such a global section does not exist in general. If we divide M into patches, we may be able to have a local section on each patch. As was demonstrated in detail in [26], it has been shown that 1. One can pick a dual triangulation Υ of M such that the map ϕ : X → M has a local section s α over each of the codimension 0 polyhedron M α in Υ.
2. The amplitude can be given as 4.2 Q → 0 limit of the light-cone gauge amplitudes As has been shown in [20], the Feynman amplitude (2.21) is well-defined for Q 2 > 10 and we can define A N (Q). If the limit Q → 0 can be taken without encountering any divergences, we will obtain the amplitude without the dilaton background. We would like to show that the limit Q → 0 is smooth if there are no divergences coming from the boundaries of the moduli space.
As has been shown in section 3, the amplitude (2.21) can be rewritten into a BRST invariant form (3.9). Since the worldsheet theory used in (3.9) consists of the matter superconformal field theory withĉ = 10 and the superconformal ghosts, the amplitude (3.9) can be recast into the form described in the previous subsection as where ω n (m; s(m)) = dm 1 ∧ dm 2 ∧ · · · ∧ dm 6g−6+2N The section s(m) here corresponds to the prescription where the PCO's are located at the interaction points of the light-cone diagram. This expression is well-defined for Q 2 > 10 but may suffer from the spurious singularities otherwise.
It is also possible to define amplitudes with the same worldsheet theory and the vertex operators, avoiding the spurious singularities by the Sen-Witten prescription. Namely we can define (0) is well-defined. We have shown that the Feynman amplitudes calculated by our method coincide with those obtained by the first-quantized method using the Sen-Witten prescription, as long as we consider infrared safe quantities.

Conclusions and discussions
In this paper, we have studied the regularization of the contact term divergences of multiloop scattering amplitudes in the light-cone gauge superstring field theory. We have used the In order to make our regularization scheme complete, we need to deal with the amplitudes for odd spin structure and those with external lines in the Ramond sector. Contrary to the dimensional regularization in which the number of the transverse dimensions d − 2 is naively shifted, it remains 8 in the present case. This implies that the present procedure does not give rise to the problem pointed out in [11] in constructing the space-time fermions. Furthermore, the light-cone gauge theory in the linear dilaton background used in this paper is much simpler than the theory proposed in [12]. We will investigate this extension elsewhere.
With our prescription, we may be able to describe superstring theory by the simple action with only three string vertices. However there exists subtle points in such a formulation. construct such a theory. We hope that we will also study these issues elsewhere.

A Arakelov metric and Arakelov Green's function
The Arakelov metricĝ A zz and the Arakelov Green's function G A (z; w) are defined as follows. Let µ zz be We note that The Arakelov metric on Σ, is defined so that its scalar curvature R A ≡ −2g Azz ∂∂ ln g A zz satisfies This condition determines g A zz only up to an overall constant, which will be chosen later. The Arakelov Green's function G A (z,z; w,w) with respect to the Arakelov metric is One can obtain a more explicit form of G A (z,z; w,w) by solving (A.6) for G A (z,z; w,w).
which can be given by Putting (A.7) and (A.5) together, we find that G A (z,z; w,w) is given by up to an additive constant independent of z,z and w,w. This possible additive constant can be absorbed into the ambiguity in the overall constant of g A zz mentioned above. It is required that (A.9) holds exactly as it is [34,37,38]. This implies that B Supersymmetric X ± CFT The conformal gauge worldsheet theory corresponding to the light-cone gauge superstring theory in noncritical dimensions was studied in [9]. The longitudinal part of it is called the supersymmetric X ± CFT whose action is given by Here the supercoordinate z is given by the superfield X ± is defined as The interaction term Γ super is given by which is the super Liouville action defined for variable Φ with the background metric ds 2 = 2ĝ zz dzdz. The super energy-momentum tensor T X ± (z) becomes where S(z, X + L ) denotes the super Schwarzian derivative In the present context, for transverse sector we use the superstring theory in a linear dilaton background instead of the theory in noncritical dimensions. We therefore consider the theory (B.1) with the identification The correlation functions to be considered in this theory are defined as Here In order to discuss these correlation functions, it is convenient to introduce supersymmetric version of ρ(z) in (2.30) which is defined by and S δ (z, z ′ ) is the fermion's Green's function corresponding to the spin structure δ. When all the external lines are in the NS-NS sector and δ is an even spin structure, S δ (z, w) is equal to the so-called Szego kernel where ϑ[δ](ζ|Ω) denotes the theta function with characteristics [δ] = δ ′ δ ′′ , namely δ = Ωδ ′ +δ ′′ (δ ′ , δ ′′ ∈ R g ), given by , (B.14) for ζ ∈ C g /(Z g + ΩZ g ) . The right hand side of (B.8) can be calculated to be Substituting (B.19) into (B.18) and differentiating with respect to Θ r ,Θ r , η s ,η s , we obtain the following identity Namely, the fermionic part of the correlation functions of supersymmetric X ± CFT coincide with those of the theory with interaction S int which is localized at the interaction points z I .

C Ghost systems on higher genus Riemann surfaces
In this appendix, we would like to show some identities which are crucial in deriving the conformal gauge expression of the light-cone gauge amplitudes in section 3.
Let us consider the conformal field theory with the action where the fields b, c are with conformal weight (λ, 0) , (1 − λ, 0) andb,c are their antiholomorphic counterparts with conformal weight (0, λ), (0, 1 − λ). Here we consider the case either λ ∈ Z or λ ∈ Z + 1 2 . The fields can be either Grassmann odd or even accordingly. We define ǫ = ±1 to be There exist local operators e qφ (z,z) q ∈ Z 2 , which satisfy We would like to discuss the correlation functions of the form on a genus g Riemann surface. q i should satisfy When λ ∈ Z + 1 2 , the correlation function we consider here is the one corresponding to a spin structure α ′ α ′′ . Namely, the fields b(z), c(z),b(z),c(z) transform as if z is moved around the a µ (µ = 1, · · · , g) cycle once, and they transform as if z is moved around the b µ cycle once. We take α ′ = α ′′ = 0 for λ ∈ Z.
Taking the metric on the worldsheet to be the Arakelov metric g A z izi , the correlation and the characteristics a b is defined so that Here ∆ is the Riemann class, which is related to the canonical divisor K of the Riemann surface by 2△ = K.

C.1 A formula for the superghosts
Since the correlation function (C.8) is left-right symmetric, we need some more work to get a formula which is useful for superghosts. We will present it in a form factorized in chiral and anti-chiral parts. By doing so, it is possible to get the correlation functions which are not left-right symmetric with respect to the choice of local operators e qφ and spin structure.
Substituting (A.8), (A.9) and into the right hand side of (C.8), we obtain for g = 1 Using (A.5), we can see Therefore there exists a holomorphic g 2 form σ (z) such that where S is independent of z. σ (z) has no zeros or poles, and it should transform as σ (z) → e −2πi △ (g−1)z ωµ+πi(g−1)Ωµµ σ (z) , (C. 16) when z is moved around the b µ cycle once, and invariant when z is moved around the a µ cycles. These properties fix σ (z) and it should coincide with the σ (z) in [30,33] up to a multiplicative factor. Substituting (C.15) into (C.13), we obtain For g = 1, instead of (C.13) we get with S, A independent of z, we get (C.17).
In (C.17), the correlation function factorizes into the left and right parts except for the determinant factor and e −3(2λ−1) 2 S . The determinant factor can also be recast into a factorized form as follows. Let us consider the bc-system with λ = 1, ǫ = 1. For arbitrary R, z i (i = 1, . . . , g), (C.17) implies and we get Therefore (C.17) can be rewritten as The correlation function dβdβdγdγ e −S βγ I |δ (β (z I ))| 2 r |δ (γ (Z r ))| 2 of the superreparametrization ghost with spin structure [α L ] for left and [α R ] for right can be evaluated to be Since Z r (r = 1, . . . N) and z I (I = 1, . . . , 2g − 2 + N) are the zeros and the poles of the meromorphic one-form ∂ρ (z) dz respectively, holds in the divisor sense. Therefore we obtain (C.26) On the other hand, (A.8), (A.9) and (C.15) imply This identity is crucial for deriving the BRST invariant expression of the superstring amplitudes.

C.2 A formula for the reparametrization ghosts
In [13] it was shown that the following identity holds: Here S bc is the action for the reparametrization ghosts, µ K (K = 1, . . . , 6g − 6 + 2N) denote the Beltrami differentials for the moduli parameters T, α, θ, and const. indicates a constant independent of the moduli parameters. The antighost insertion dz ∧ dz i µ K b +μ Kb corresponding to the variations of T, α, θ are given by the following contour integrals: • The stretch corresponds to the variation T → T + δT of the height T of cylinders. Let us order the interaction points z I (I = 1, . . . , 2g − 2 + N) so that Reρ(z 1 ) ≤ Reρ(z 2 ) ≤ · · · ≤ Reρ(z 2g−2+N ) , (C. 30) and define the moduli parameters corresponding to the heights as The antighost insertion corresponding to the deformation T I → T I + δT I is given by where C i denotes the contour around a cylinder which includes the region Reρ(z I ) ≤ Reρ ≤ Reρ(z I+1 ) and the sum should be taken over all such contours. (See figure 5.) There are 2g − 3 + N insertions of this kind.
where C twist is the contour around the cylinder which is twisted. There are 3g − 3 + N insertions of this kind.
• The shift corresponds to the variation of the loop momenta preserving the momenta of the external lines. The antighost insertion for such a variation becomes where C shift is taken as in figure 6. There are g insertions of this kind.
Here ε K = −1 for the stretches and ε K = 1 for the twists and shifts.
D A proof of (3.9) We would like to prove (3.9) by showing that the right hand side of (3.9) is equal to that of (3.8). We consider the generic situation in which z I , Z r are all distinct.
What we do first is to rewrite the PCO's X (z I ) ,X (z I ) using the existence of nilpotent fermionic chargeQ One can show that X (z I ) can be expressed as andQ satisfies the following identities: are both equal to the total momentum in the + direction through the channel which is fixed by the external momenta, the right hand side of (D.5) should vanish. In the case of the contour integrals (C.33), (C.34), we obtain which vanishes because X + and ρ +ρ should be singlevalued. HenceQ commutes or anticommutes with all the quantities in (3.11). Therefore the second term on the right hand side of (D.2) does not contribute to the correlation functions, because it isQ exact. We can replace all the X (z I ) in the correlation functions by and similarly forX (z I ), the right hand side of (3.9) becomes a sum of the right hand side of (3.8) and the terms which involve which appear in (3.13) and (3.14), in place of e φ T LC F (z I ) , eφT LC F (z I ). The X ± CFT part of such terms are of the form O − (z I 1 ) · · · O − (z In ) O + z I n+1 · · · O + z I n+m Ō − · · ·Ō + · · · × contributions from S,S, V DDF with z J = z I , because S int is localized at the interaction points and involves even number of ψ + 's.
Thus we have shown that the terms of the form (D.11) all vanish and the right hand side of (3.9) is equal to that of (3.8).