S-duality improved perturbation theory in compactified type I/heterotic string theory

We study the mass of the stable non-BPS state in type I/heterotic string theory compactified on a circle with the help of the interpolation formula between weak and strong coupling results. Comparison between the results at different orders indicate that this procedure can determine the mass of the particle to within 10% accuracy over the entire two dimensional moduli space parametrized by the string coupling and the radius of compactification. This allows us to estimate the region of the stability of the particle in this two dimensional moduli space. Outside this region the particle is unstable against decay into three BPS states carrying the same total charge as the original state. We discuss generalization of this analysis to compactification on higher dimensional tori.


Introduction
Our current understanding of string theory is based mostly on perturbation expansion in the string coupling [1]. Furthermore this perturbation expansion is believed to be an asymptotic expansion. For this reason one might worry that our ability to compute anything in string theory may be limited to very narrow corners of the full string theory landscape -regions in which the theory admits a description as a very weakly coupled string/M/F-theory.
Ref. [2] suggested making use of duality and suitable interpolation formula to translate the weak coupling results in string theory to approximate results for physical quantities over the entire range of string coupling constant. As a specific example, the mass of the stable non-BPS particle in ten dimensional type I/SO(32) heterotic string theory was considered. Using a suitable formula that interpolates between the result for this mass in weakly coupled SO(32) heterotic string theory and weakly coupled type I string theory, an approximate formula for the mass of this state was derived over the entire range of string coupling. Furthermore by comparing the results in different orders it was estimated that this approximate formula lies within 10% of the exact result over the entire range of coupling. Generalization of this analysis has since been discussed in [3][4][5].
Since a generic string theory moduli space is multi-dimensional it is natural to ask if this interpolation technique can be used to find approximate expressions for various physical quantities over the full multi-dimensional moduli space. 1 In this paper we explore JHEP06(2014)068 this in the context of SO(32) heterotic/type I string theory compactified on a circle. If we do not switch on any Wilson line so that the SO(32) gauge group is unbroken then the moduli space is two dimensional, parametrized by the string coupling and the radius of compactification. We use perturbative results in the compactified string theory and a suitable generalization of the interpolation formula used in [2] to derive expression for the mass of the non-BPS state in the full two dimensional moduli space. Comparison between different orders of approximation again indicates that the approximate formula derived here lies within 10% of the exact formula over the entire two dimensional moduli space.
In type I/SO(32) heterotic string theory compactified on S 1 , one can identify a set of BPS states whose total charge is the same as that of the charge carried by the non-BPS state under study. Thus the latter can decay into the former if the mass of the non-BPS state is larger than the sum of the masses of the BPS states to which it could possibly decay. With the help of the approximate formula for the mass we determine the part of the region of the two dimensional moduli space in which the non-BPS state is unstable. Again we find that the region determined this way is only mildly sensitve to the order of the approximation that we use.
The rest of the paper is organized as follows. In section 2 we review the interpolation technique and the normalization conventions of [2], and the additional ingredients we need for dealing with the compactified theory. In section 3 we carry out the computation of the one loop correction to the mass of stable non-BPS state in type I string theory compactified on S 1 . In section 4 we carry out a similar calculation in SO(32) heterotic string theory compactified on S 1 . In section 5 we construct the interpolation formula for the mass of the non-BPS state in various approximation, and compare the results in different orders of approximation. In section 6 we use the interpolation formula to analyze the region of stability of the non-BPS state. In section 7 we discuss extension of our analysis to the case of compactification on higher dimensional tori. We conclude in section 8 with a brief summary of our results and their possible relation to related developments.

Normalization conventions and tree level results
We shall use the normalization conventions used in [2]. The purpose of this section will be to review these conventions and also introduce the extra conventions involving the radius of compactification.
Let g H and g I be the string coupling in ten dimensional heterotic and type I string theories respectively. We introduce a new coupling parameter g in terms of which g H and g I are given by g H = 2 7/2 π 7/2 g, g I = 2 3/2 π 7/2 g −1 . (2.1) We normalize the heterotic and type I metric such that the heterotic string tension in heterotic metric and the type I string tension in type I metric are both given by 1/2π. The mass of the non-BPS state, measured in the ten dimensional Einstein metric, is parametrized by

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The tree level weak and strong coupling values of F (g), computed respectively from tree level heterotic and type I string theories, are Upon compactification on a circle the tree level masses will continue to be given by (2.3) if we measure it in the canonical metric in ten dimensions. We shall follow this convention. Let r I , r H and r E denote the radii of the compact circle measured in the type I, heterotic and canonical metric. Then r E = (g I ) −1/4 r I = 2 −3/8 π −7/8 g 1/4 r I , r H = (g H ) 1/4 r E = 2 7/8 π 7/8 g 1/4 r E = 2 1/2 g 1/2 r I . (2.4) We expect the quantum corrections in the heterotic and type I string theories to modify the weak coupling results to where the functions A 2k (r H ) and B n (r I ) have finite r H → ∞ and r I → ∞ limits respectively, corresponding to the results in the non-compact theory. We now introduce the interpolating function 2 F m,n (g) = g 1/4 1 + a 1 (r H )g + · · · a m (r H )g m + b n (r I )g m+1 + b n−1 (r I )g m+2 + · · · + b 1 (r I )g m+n + g m+n+1 where the functions a k (r H ) and b k (r I ) are determined as follows. We determine a k (r H ) by demanding that after setting the b k 's to zero, the expansion of (2.6) in powers of g at fixed r H agrees with that of F W (g) up to order g 1 4 +m . Similarly the functions b k (r I ) are determined by demanding that after setting the a k (r H )'s to zero, the expansion of (2.6) in powers of g −1 at fixed r I agrees with that of F S (g) up to order g 3 4 −n .
We shall now argue that the weak coupling expansion at fixed r H of the full function F m,n keeping both a k 's and b k 's non-zero coincides with that of F W up to order g 1 4 +m , and similary the strong coupling expansion at fixed r I of the full function F m,n coincides with that of F S up to order g 3 4 −n . From eq. (2.4) it follows that r I = 2 −1/2 g −1/2 r H , and hence as g → 0 keeping r H fixed, r I → ∞. In this limit the coefficients B k appearing 2 There are various other possible interpolation schemes (see e.g. [6,7]), but the one given in (2.6), called the fractional power of polynomial (FPP) scheme in [3], seems to be most suitable for our purpose as this gives a clear separation between the coefficients which are determined using weak coupling expansion and the coefficients which are determined using strong coupling expansion. This is needed to ensure that the weak coupling expansions at fixed rH and strong coupling expansion at fixed rI match the perturbation expansions. The difficulty in achieving this with other approximation schemes, e.g. 2-point Padé approximant, is similar to the difficulties faced in [3] in getting a duality invariant approximation scheme beyond four loops using 2-point Padé approximant.

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in the strong coupling expsnsion should approach finite values given by the results in the non-compact theory. Thus the coefficients b k , determined in terms of the coefficients B for k ≤ n, should also approach finite values in this limit. This shows that the expansion of b k in powers of g at fixed r H contains non-negative powers of g. Substituting this into (2.6) we now see that the coefficents b k do not affect the weak coupling expansion of F m,n to order g 1 4 +m , and hence the weak coupling expansion of F m,n to this order agrees with that of F W . A similar analysis shows that the expansion of a k in powers of g −1 at fixed r I contains non-positive powers of g. Hence the expansion of F m,n in powers of g −1 at fixed r I to order g 3 4 −n is insensitive to the coefficients a k and coincides with that of F S (g). From (2.3), (2.6) we can find the following interpolating functions for the mass of the non-BPS particle

Strong coupling expansion
Denote by ∆M the first order correction to the mass formula from the strong coupling end, i.e. in weakly coupled type I string theory compactified on a circle S 1 of radius r I . This can be obtained by calculating the one loop correction to the energy of the non-BPS D0-brane of type I string theory compactified on S 1 . This calculation differs from the corresponding calculation in [2] by having to include extra contribution from open string winding modes along the circle, begining on a D0-brane and ending on one of its images. The result takes the form [8][9][10][11] − ∆M = 1 2 g where Z N S;D0D0 , Z R;D0D0 , Z N S;D0D9 , Z R;D0D9 denote respectievely the contributions from the NS and R sector open strings with both ends on the D0-brane and NS and R open strings with one end on the D0-brane and the other end on the D9-brane wrapped on S 1 of radius r I . Explicit computation gives

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where n is the quantum number corresponding to the winding number of the fundamental open string along the compact direction and Individual terms in (3.1) are both IR and UV divergent. Using the prescription for the IR and UV regularization described in [2] we can express (3.1) as follows, Note that for r I < 1/ √ 2, the n = 1 term in the sum behaves as q 2r 2 I −1 = e πs(1−2r 2 I ) and hence the integral over s has a divergence from the large s region. This reflects the appearance of the open string tachyon in the spectrum for r I < 1/ √ 2 [12]. For this reason the open string loop corrections to the mass of stable non-BPS state makes sense only for r I ≥ 1/ √ 2, and in the rest of this section we shall focus on this region. Using (2.4) we see that in terms of the radius r H in the heterotic metric, this condition takes the form It is possible to convert expression (3.6) in the 'closed string channel' using the modular tranformation laws of f i 's:

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(3.9) C 00 denotes the cylinder amplitude with boundaries lying on the D0-brane, given by the inner product between the boundary states of D0-brane. C 09 denotes the cylinder amplitude with one boundary lying on the D0-brane and the other boundary on the D9-brane wrapped on S 1 , given by the inner product between the boundary states of D0-brane and the D9-brane wrapped on S 1 . M denotes the möbius strip amplitude with boundary lying on the D0-brane, given by the inner product between the boundary states of D0-brane and the crosscap. Using this and eqs. (2.1), (2.2) we can write the corrected strong coupling expression for F S (g, r I ) to order g 3 4 −1 as, (3.10) K S (r I ) can be obtained by integrating expression (3.6) numerically for different values of r I . We find that the result of this numerical evaluation fits well with the function within 1% accuracy over the entire range 1/ √ 2 ≤ r I < ∞.

Weak coupling expansion
First order correction to M in the weakly coupled heterotic string theory given by where K W (r H ) can be calculated by doing a one loop heterotic string calculation similar to that in [2], but including the effect of closed heterotic string winding and momentum modes along the circle. The result is with τ denoting the modular parameter of the torus, ν denoting the spin structure on the torus taking values 00, 01, 10 and 11, ϑ are the Jacobi theta functions, r H radius of S 1 on which heterotic string theory is compactified and n, w representing the momentum and winding number along the compactified direction. Since this expression is invariant under

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T-duality transformation r H → 1/r H (except for the overall factor of 1/r H that is taken care of by the transformation law of g 2 multiplying it), we can focus on the region r H ≥ 1.
In this region the evaluation of the integral can be facilitated using a Poisson resummation in the variable n. This yields In the r H → ∞ limit only k = w = 0 term in the sum survives, giving back the ten dimensional result. .

(4.4)
Then the corrected weak coupling expression for F W (g, r H ) to order g 2 is given as, Notice that K W (r H ) diverges in the r H → 0 limit. This is easily understood using the known T-duality invariance r H → 1/r H in the heterotic string theory. Under this the ten dimensional string coupling transforms to g/r H . Defining we can express (4.5) as Except for the overall factor of ( r H ) −1/4 which reflects the overall scale factor relating the ten dimensional Einstein metric in the dual pair of heterotic string theories, we see that this has a perfectly good r H → ∞ (r H → 0) limit at fixed g. For this reason, for r H < 1 it is more natural to use the coupling constant g of the T-dual theory as an expansion parameter.

Interpolating functions
We now turn to the construction of the interpolating functions. For definiteness we shall treat g and r H as independent variables. We shall divide up the g-r H plane into several regions shown in figure 1 and use different interpolating functions in these different regions. Region I: first consider the region I defined by In this region r I ≥ 1/ √ 2 (see (3.7)) and both the heterotic perturation theory in powers of g and type I perturbation theory in powers of g −1 are well defined at small and large g respectively. Thus we can use standard interpolation formula described in section 2: F 0,1 (g, r H ) = g 1/4 (1 + 4 K S (r I )g + g 2 ) 1/4 , (5.4) F 1,1 (g, r H ) = g 1/4 (1 + 6 K S (r I )g 2 + g 3 ) 1/6 , (5.5) F 2,0 (g, r H ) = g 1/4 (1 + 6 K W (r H )g 2 + g 3 ) 1/6 , (5.6) F 2,1 (g, r H ) = g 1/4 (1 + 8 K W (r H )g 2 + 8K S (r I )g 3 + g 4 ) 1/8 , (5.7) F 3,0 (g, r H ) = g 1/4 (1 + 8 K W (r H )g 2 + g 4 ) 1/8 , (5.8) and F 3,1 (g, r H ) = g 1/4 1 + 10 K W (r H )g 2 + 10 K S (r I )g 4 + g 5 1/10 . (5.9) Region II: region II is defined by In this region heterotic perturbation theory in power of g is still valid for small g but type I perturbation theory in powers of g −1 breaks down at large g due to the presence of the tachyon. Thus the only interpolating functions we can use are F m,0 for 0 ≤ m ≤ 3.
Region III: region III is defined by

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The significance of this region can be understood by reexpressing (5.11) in terms of T-dual variables r H , g introduced in (4.6). This corresponds to In this region we shall use the interpolating functons of region I with (g, r H ) replaced by ( g, r H ) and with an overall multiplicative factor of ( r H ) −1/4 = (r H ) 1/4 to account for the rescaling of the canonical metric discussed below (4.7). Thus we use the functions Physically this corresponds to using an interpolating formula between the T-dual heterotic string theory and its strong coupling dual type I string theory (obtained in the g → ∞ limit at fixed r H / g 1/2 ). This is not the original type I string theory, but related to it via a strong-weak coupling duality transformation. This is apparent from the fact that while in the original type I string theory the non-BPS D-brane develops a tachyon for r H < g 1/2 , in the new theory the non-BPS D-brane is tachyon free for r H < g −1 .
Region IV: region IV is defined by In the ( g, r H ) variables this corresponds to 1 ≤ r H ≤ g −1/2 , i.e. this is the heterotic T-dual image of region II. Thus we use the interpolation formulae of region II with (g, r H ) replaced by ( g, r H ): Note that the results in regions III and IV can be obtained from those in regions I and II by heterotic T-duality transformation (4.6). For this reason we shall focus on regions I and II from now on.
In figure 2 we have plotted the ratios of F m,n to F 3,1 as a function of g for four different values of r H in region I. As we can see, except for F 0,0 , all other F m,n 's remain within about 10% of F 3,1 over the entire allowed range of g in region I. This suggests that F 3,1 gives the actual mass of the particle within about 10% error over the entire range of parameter space of region I (and hence also of region III). We shall return to a discussion of regions II and IV in the next section.  After taking into account the normalization (2.2) we get This expression is not renormalized. Furthermore it is manifestly invariant under heterotic T-duality transformation (4.6). Perturbative stability of non-BPS states in compactified SO(32) heterotic string theory has been analyzed in [13]. We shall now analyze non-perturbative stability of these states using the interpolation formula. In the (m, n) approximation the non-BPS particle is stable when its mass is less than the total mass of the BPS constituents to which it can decay. In regions I and II this requires F m,n (g) < g 1/4 r H + 1 r H . (6.4) In regions III and IV the left hand side is replaced by F m,n (g), but the results in these regions are related to those in regions I and II respectively by heterotic T-duality. Now one can check explicitly that in region I all the F m,n (g)'s satisfy (6.4), showing that whatever approximation we use, the non-BPS state is stable in this region. By heterotic T-duality the same result holds in region III. In region II only F m,0 approximations make sense. In figure 3 we have shown by the shaded region the region of instability of the non-BPS state in different approximations. As we can see, these regions are not too different from each other, indicating that this is a fairly good approximation to the true region of instabiity of the non-BPS state in region II. The region of instability in region III can be found from this using heterotic T-duality transformation. One point worth noticing is that in each of these plots, there is a narrow strip of region II where the non-BPS state is stable. This may seem a bit surprising at first since in the whole of region II perturbative open string theory describing the non-BPS D0-brane develops a tachyon. Note however that this is true only in tree level open string theory which corresponds to g → ∞ limit of this diagram. Indeed in this limit the strip width reduces to zero showing that the D0-brane becomes unstable as soon as we cross the upper boundary of region II. At finite g however the tachyon mass 2 itself may get corrected and hence the tachyon may not develop as soon as we cross the upper boundary. We cannot do this analysis directly in type I string theory since at present it is not understood how to carry out open string perturbation theory in the presence of a tree level tachyon. Instead we have checked the stability by comparing the mass of the unstable brane with the total mass of the decay products, and arrived at figure 3.

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It is natural to ask whether one can reliably determine the mass of the non-BPS particle in the white part of region II where it is expected to be stable. This can be done via the interpolating function F m,0 . Figure 4 shows the ratios of F m,0 to F 3,0 in region II. As we can see from this graph, the ratios remain within about 10% of unity except for F 0,0 , indicating that the interpolating formulae based on F 3,0 determines the actual mass of the stable non-BPS particle even inside region II to within about 10% accuracy.

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for any non-zero n. Inside this region we need to use only the zeroeth order result on the strong coupling side. On the weak coupling side, when the size of the torus T d is small, K W computed from (7.3) becomes large signalling an apparent breakdown of perturbation theory. The remedy is to use a T-duality transformation and use the T-dual variables. In fact here we have a large group O(6, 6; Z) of T-duality transformation acting on the moduli space. We need to identify the analog of the regions I and II in figure 1 in which we carry out the actual computation and interpolation and then extend the result to the rest of the moduli space using heterotic T-duality invarinace. Since in the compactified heterotic string theory the effective coupling constant is given by g 2 / √ det G H , the natural analog of regions I and II will be to pick that domain in the moduli space for which √ det G H takes the maximum possible value -i.e. given any point inside such a domain, any of its T-dual image should have lower value of √ det G H . Once we have identified such a domain we then divide this into the two regions I and II depending on whether 2G Ik n k n lies above 1 for all n or not. The rest of the analysis would then proceed as in the case of S 1 compactification.

Conclusion
In this paper we have analyzed the mass formula for stable non-BPS state in type I/SO(32) heterotic string theory compactified on a circle using the interpolation formula between the strong and weak coupling results. Our analysis indicates that the interpolation formula determines the mass of the state within 10% accuracy over the entire moduli space. We also determine the region of stability of the particle based on the mass formula and discuss generalization of the analysis for generic toroidal compactification.
In recent times there has been significant developments in resumming perturbation theory [14][15][16][17][18][19][20][21][22][23][24][25]. It will be interesting to see if the interpolation between strong and weak coupling results can be combined efficiently with these resummation techniques to get a better understanding of physical quantities at intermediate values of coupling.