Superpotential de-sequestering in string models
Non-perturbative superpotential cross-couplings between visible sector matter and Kähler moduli can lead to significant flavour-changing neutral currents in compactifications of type IIB string theory. Here, we compute corrections to Yukawa couplings in orbifold models with chiral matter localised on D3-branes and non-perturbative effects on distant D7-branes. By evaluating a threshold correction to the D7-brane gauge coupling, we determine conditions under which the non-perturbative corrections to the Yukawa couplings appear. The flavour structure of the induced Yukawa coupling generically fails to be aligned with the tree-level flavour structure. We check our results by also evaluating a correlation function of two D7-brane gauginos and a D3-brane Yukawa coupling. Finally, by calculating a string amplitude between hidden scalars and visible matter we show how non-vanishing vacuum expectation values of distant D7-brane scalars, if present, may correct visible Yukawa couplings with a flavour structure that differs from the tree-level flavour structure.
Department of Physics, Karlstad University
651 88 Karlstad, Sweden
Oskar Klein Center, Stockholm University
Albanova University Center
106 91 Stockholm, Sweden Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Department of Physics,
Cornell University, Ithaca, NY 14853 Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
Understanding the effects of moduli stabilisation for semi-realistic models of particle physics arising from string theory is an important and challenging question in string phenomenology. By localising the visible sector geometrically in the extra dimensions, one may hope for a decoupling of global effects and thereby a simplified framework for extracting the predictions of the model (e.g. 0005067 (); Verlinde:2005jr ()). A number of obstructions to such simplifications have been discussed in the literature, and in this note we focus on one specific example of an effect induced by moduli stabilisation that can alter the visible sector spectrum.
Sequestering Randall:1998uk (), a strategy to surmount the famous “supersymmetry flavour problem”, is one example of a desirable property that is supposed to follow from localising the visible sector geometrically. Specifically, it was argued in Randall:1998uk () that for a five-dimensional model where only gravity propagates in the extra dimension and in which the visible and hidden sectors are geometrically separated, the gravity mediated soft terms vanish at tree-level Randall:1998uk (). The low-energy soft-terms were instead argued to be flavour universal and to arise first at loop level through anomaly mediation.
It may seem that this attractive mechanism should arise naturally in string theory models in which the visible and hidden sectors are realised as branes filling four-dimensional space-time while being geometrically separated in the internal compactification manifold. However, in Anisimov:2001zz (); Anisimov:2002az () it was shown that for a variety of string models in Type II string theory and M-theory, sequestering does not arise naturally. The discrepancy between these two results can be understood as a consequence of Kaluza-Klein (KK) modes with masses of the order of the compactification scale, and moduli which only become stabilised after compactification and which therefore have masses below the compactification scale Anisimov:2001zz (); Kachru:2006em (). These fields typically mediate interactions which spoil sequestering and re-introduce the supersymmetry flavour problem in these models.
By localising the supersymmetry breaking sector at the bottom of a warped throat, sequestering may still occur in large classes of string compactifications with flux Kachru:2007xp (); Luty:2000ec (), consistently with the expectations from the conformal field theory (CFT) dual, known as “conformal sequestering” Luty:2001jh (). This type of models can be constructed in Type IIB flux compactifications Giddings:2001yu (), for example.
Furthermore, effective theories which are not of sequestered form but satisfy the milder criterion of “sort-of-sequestering” Berg:2010ha () still share with fully sequestered models the favourable avoidance of the supersymmetry flavour problem at tree level. Interestingly, compactifications in the Large Volume Scenario (LVS) Balasubramanian:2005zx (); Conlon:2005ki (), are of this form Berg:2010ha (), due to “extended no-scale structure” Cicoli:2007xp ().
However, taking the dynamics of moduli stabilisation into account creates another adverse effect, where even warping or “sort-of-sequestering” may not suffice to shield the visible sector from phenomenologically dangerous contributions to the soft terms. This effect, which intimately couples the low-energy phenomenology of the localised visible sector to the details of the moduli stabilising sector, was called superpotential de-sequestering in Berg:2010ha ().
This effect is particularly important in compactification schemes in which there is a hierarchy between the gravitino mass and the scale of the soft masses, and is of particular interest in Kachru-Kallosh-Linde-Trivedi (KKLT) type models Giddings:2001yu (); Kachru:2003aw () and in the Large Volume Scenario Balasubramanian:2005zx (); Conlon:2005ki (), as we will now review.
Non-perturbative effects in the form of Euclidean D3-branes or gaugino condensation on a stack of D7-branes wrapping some four-cycle in the internal dimensions are an essential ingredient in several celebrated moduli-stabilisation schemes in type IIB string theory. At the level of the four-dimensional effective theory, the non-perturbative effects induce a non-perturbative superpotential which depends on the volume of the corresponding four-cycle. Once supersymmetry is broken, non-vanishing F-terms can be induced for the Kähler moduli. Superpotential de-sequestering refers to interactions induced between the Kähler moduli and a localised visible sector which is geometrically separated from the four-cycle supporting the non-perturbative effects. In particular, the presence of operators of the form,
for some visible sector operator and for some Kähler modulus , can give rise to important contributions to the soft terms in certain moduli stabilisation scenarios. We immediately note that at first sight 1 may seem negligible since it is a nonperturbative contribution, but since analogous nonperturbative contributions are also of crucial importance in stabilising the moduli in scenarios under consideration, this argument needs careful scrutiny. There turn out to be numerically important contributions, and the most important contributions are those to the -term and the soft -terms involving the Higgs field and leptons and squarks.111There are essentially no bounds on -terms that do not involve the Higgs.
In this note, we will focus specifically on superpotential operators of the form,
where is some constant and the are various charged fields. After supersymmetry breaking in which the Kähler modulus obtains a non-vanishing -term, the induced -terms are given (in supergravity) by
in natural units, where are the full, moduli-dependent superpotential Yukawa couplings. Note that this expression applies to non-canonically normalised fields, but can easily be extended to canonically normalised fields. (Even if the visible sector fields come with non-diagonal Kähler metric, one can still apply general coordinate transformations in field space to extend the expression to canonically normalised fields.) If one of the fields in equation (2) is a Higgs field, soft masses for the remaining fields will be induced after electroweak symmetry breaking. These masses can be bounded by studying their effect on flavour-violating and CP-violating processes in the standard model. The consequences of the bound thus obtained depend on the moduli stabilisation scenario, and the size and structure of . In particular, if , much weaker bounds apply than would otherwise be the case.
Computing the de-sequestering operators
In this paper, we study de-sequestering in some simple toroidal orbifolds which serve as computable toy models for more complicated compactifications. We model the visible sector as arising from D3-branes at an orbifold singularity, while the D7-branes supporting the non-perturbative effects can either wrap two 2-tori of the compact space or a small 4-cycle instead. In the orbifold limit, D7-branes wrapping a small cycle become D3 branes at orbifold singularities. In our study we are motivated by previous results of flavour-violating effects due to Yukawa couplings sourced by string or D-brane instantons Abel:2006yk (); Buican:2008qe (); Cvetic:2009yh (). Similarly, nonperturbative effects in F-theory GUT models have been found to modify the flavour structure of the Yukawa couplings Marchesano:2009rz ().
For non-perturbative effects arising from gaugino condensation on a stack of D7-branes, the effective superpotential is proportional to
where is the gauge kinetic function on the D7-branes, and , where is the dual Coxeter number of the D7-brane gauge group, so that e.g. for . The dependence of the D7-brane gauge kinetic function on the visible sector fields thus induces operators of the form of equation (1). This will enable us to compute the contribution to the physical -terms from superpotential de-sequestering by world-sheet techniques. Let us outline what couplings we need to compute.
In the low-energy effective field theory the quantity of interest contributes to the D7-brane gauge kinetic term of the SUSY Lagrangian:
where is the holomorphic gauge kinetic function depending on D3 matter fields and moduli , whereas is the D7 field strength superfield. We can expand the gauge kinetic function in gauge-invariant combinations of the D3 matter fields :
It is the trilinear term that will be responsible for generating superpotential terms of the form (2), so we want to determine . After integrating over the Grassmann coordinates , the term will produce supermultiplet component couplings
where are D7 gauge bosons, the corresponding gauginos and and are D3 fermionic and scalar matter fields. Thus there are two kinds of string amplitude calculations we could perform to obtain , and we will consider both. First, we compute the one-loop threshold correction to the D7-brane gauge kinetic function given by . Secondly we will calculate corrections to D3 Yukawa couplings due to gaugino condensation by evaluating . Both amplitudes are computed as a cylinder diagram with two D7-brane gauge boson or gaugino vertex operators inserted at one boundary, and three visible sector matter field operators inserted at the other boundary.
The string calculations will be performed for a range of orbifold models. We will find that only Yukawa couplings with particular flavour structures appear in the nonperturbative superpotential. Most importantly their flavour structure does not match that of the tree level Yukawa couplings and thus de-sequestering can introduce flavour violation.
Even though these orbifold models are just simple toy models, they capture the relevant features to determine whether superpotential de-sequestering can occur in these D-brane setups. We will therefore argue that our findings are generic for string backgrounds including D3/D7-branes and thus apply to popular compactification schemes with moduli stabilisation, such as KKLT and LVS.
While the precise expressions will be sensitive to the finer details of the model, the existence or absence of de-sequestering is governed by a few simple properties of the string theory setup. In particular, we do not need to specify the full model to determine the flavour structure of the induced operators. We do restrict attention to localising the visible sector on D3-branes at a supersymmetric singularity, but we consider it likely that D7-brane matter would give similar results.
We find that the relevant string theory backgrounds fall into three classes which are displayed in figures 1, 2 and 3. The setups differ in the realisation of the non-perturbative effects. We now analyse them in turn.
In the first configuration shown in fig. 1 the D7-branes wrap a small cycle in the compact space. This is relevant in the LVS scenario, where the small cycle can be identified as a blow-up cycle. While the D3- and D7-stacks are separated in the compact space, they are connected by a homologous 2-cycle.222In orbifold models this is equivalent to the orbifold possessing a supersymmetric sector. In this sector the stacks of branes are separated along an untwisted direction allowing for the propagation of string modes between them. We find that this situation leads to de-sequestering: the homologous 2-cycle allows for the propagation of string modes between the stacks of branes thus inducing terms in the non-perturbative superpotential.
The second setup displayed in 2 is very similar to the case considered above: the D7-branes again wrap a small cycle in the compact space. This time, however, the visible stack and the non-perturbative effects do not share a homologous 2-cycle in the compact geometry.333In this case the orbifold does not display any sectors connecting the two sectors. As mediating closed string modes can only propagate on the appropriate cycles, warping can help in KKLT Kachru:2007xp () and the aforementioned “sort-of-sequestering” can help in LVS Berg:2010ha (). We find that in this case geometric separation, together with these two effects, appears to suffice to avoid de-sequestering.
The third setup is sketched in 3 and arises in KKLT constructions where non-perturbative effects are located on bulk D7-branes wrapping a 4-cycle.444These effects are irrelevant in LVS as they are suppressed by a factor of in the superpotential. The effects of this setup are very similar to the first case considered above: while the two stacks of branes are separated in the compact space, they nevertheless communicate via closed string modes.555In the orbifold setting the relevant open string modes arise in the untwisted sector. The result is again that contributions to the non-perturbative superpotential will be generated, thus leading to de-sequestering.
The above results allow us to identify a necessary condition for de-sequestering to occur, independent of the phenomenological model: the stack of D7-branes carrying the non-perturbative effects has to either wrap a bulk 4-cycle or share a homologous 2-cycle with the stack of visible sector D3-branes.
These conditions pose further restrictions on the type of singularity at which the visible sector is realised. In case (i) above de-sequestering is absent if the singularity is too trivial as it has to share a homologous 2-cycle. On the other hand, there is no such condition in case (iii). There de-sequestering can occur for singularities as simple as (which corresponds to ).
Another important result is that the flavour structure of the induced Yukawa couplings does not match with the one of the tree-level Yukawa-interactions. This observation is general whenever there is de-sequestering, both in scenarios relevant to KKLT models and the LVS. Hence we confirm the validity of the effective field theory argument of Berg:2010ha (), that superpotential de-sequestering can lead to large flavour-changing neutral currents (FCNCs).
While the position moduli of D7-branes wrapping bulk cycles generically become supersymmetrically stabilised with the complex structure moduli at a relatively high scale Kachru:2007xp (), matter scalars for D7-branes wrapping small cycles may not. We show that the Yukawa couplings of the D3-brane visible sector obtain a dependence on fields characterising the distant D7-brane wrapping a small cycle, and the flavour structure of the resulting Yukawa couplings is different from the tree-level flavour structure.
2 The Model: Toroidal Orbifold Backgrounds
In this section we review some relevant background material on toroidal orbifolds for the backgrounds in which we intend to do the computation. Specifically, we will discuss different models for the visible sector and for the non-perturbative effects as well as the spectrum of local orbifold models.
2.1 Modelling the visible and the hidden sectors
Both the visible sector and the sector supporting the non-perturbative effects can be modeled in several different ways in toroidal orbifolds, and here we discuss the virtues and drawbacks of some of the conceivable alternatives, which are also summarised in table 1.
|Alternative||Visible sector||Non-perturbative effects||Model|
|#1||Bulk D3||Bulk D7||not used|
|#2||Bulk D3||D3 at orbifold singularity||not used|
|#3||Fractional D3||Bulk D7||“KKLT”|
|#4||Fractional D3||D3 at orbifold singularity||“LVS”|
Bulk D3-branes (i.e. D3-branes at a smooth point) support supersymmetry on the world-volume, and correspondingly the scalars of such a visible sector transform in the adjoint representation of the gauge group. This makes bulk D3-branes less than realistic as a toy model for a chiral model of particle physics and we will not discuss them any further in this work.
D3-branes at orbifold singularities give rise to ‘quiver’ theories, in which the scalars transform under the bifundamental representation of the gauge groups of some fractional branes Douglas:1996sw (). We will be more specific in the explicit examples discussed below.
In the various moduli stabilisation scenarios of interest, the non-perturbative effects can be realised either as gaugino condensation on a D7-brane on a bulk cycle or a blow-up cycle. The latter case is more interesting for applications to the Large Volume Scenario in which the non-perturbative effects are supported on a small (though still above string scale) four-cycle. The former case is interesting in e.g. the KKLT scenario. While a bulk cycle in a toroidal orbifold is realised as a D7-brane that wraps two ’s in the internal space but is pointlike in the third , a ‘small’ cycle can be modeled as a D3-brane at an orbifold singularity in the toroidal computation. Let us emphasise here that the fractional D3-brane that would model a D7-brane at a small cycle should be placed at an orbifold singularity that is invariant under the orbifold action (e.g. the origin of a orbifold), so that it does not carry twist charge.
2.2 Local vs. Global
There certainly exist arrangements of brane stacks and orientifold planes that ensure tadpole cancellation in global (compact) models such as the orientifold version of in fig. 4. (For a review of orientifolds, see Angelantonj:2002ct ().) We will argue that our calculation of the string one-loop renormalisation of a certain five-point double trace operator can be performed mostly in a local model of the singularity and should carry over mostly unchanged to a global orientifold model. What we mean by “mostly” is that certain zero-mode solutions involve states that do propagate away from the singularity, but this is under control.
The statements in the previous paragraph are not obvious, but we have two simple arguments why they are true:
a) Finiteness. As is well known, for a generic D-brane configuration, the cylinder vacuum amplitude has divergences due to long-distance propagation of massless closed string modes at zero momentum, i.e. , which can be ascribed to a tadpole diagram attached to a D-brane, i.e. a nonzero probability for a zero-momentum closed string to be produced from the vacuum. This problem is solved by imposing tadpole cancellation. In some planar open-string amplitudes (see e.g. Berg:2011ij ()) lack of enforcement of tadpole cancellation would cause problematic divergences, because factorisation onto closed string states recreates any uncancelled tadpoles on the side of the cylinder without external states inserted (see fig. 5).
This means that even if one tries to argue that tadpoles are cancelled once the model is completed to a global model, the meaning of a calculation that suffers from this problem, but is actually performed only in the local model where there remain uncancelled tadpoles, is unclear.
However, in the spirit of Dudas:2004nd (), certain calculations do not suffer from this problem. In a cylinder amplitude with insertions on both sides, the situation is generically better, because there is momentum flowing in the propagator , and hence no divergence. One expects poles from reducible diagrams, but this is perfectly physical. Still, in some of our examples we are interested in a potential term, which has no derivatives, so we want to take the external momenta on the scalar side to zero. One could worry that the problem would reappear, but calculation shows it does not (similar arguments recently appeared in Anastasopoulos:2011kr ()). In this limit, the amplitude behaves as (see e.g. 147 below).
From the field theory point of view, this is a nontrivial physical statement about reducible diagrams, that the vertices each supply a factor of momentum such that the overall diagram is finite in the massless and low-energy limit. In string theory, this instead reflects something technical but essentially trivial, that if we begin with something finite with no momentum-dependent forefactors, then change picture (in the sense of superconformal ghost charge, see below) in a way that produces some explicit additional momenta in a given amplitude, we must also produce compensating poles of the kind above, so as to not change the final result.
b) Double trace. As noted above, if all vertex operators had been inserted on one side of a cylinder diagram as in fig. 5, then there could have been a potential divergence and it would have been necessary to compute the corresponding Möbius strip diagram. However, the interactions we are interested in are double trace operators. This requires two boundaries on the worldsheet, and at one loop the only topology that will allow this is the cylinder, not the Möbius strip. In other words, we cannot insert vertex operators on an orientifold plane (it has no dynamics of its own), so there is no analog of the Möbius strip amplitude for the couplings we are interested in. Therefore, at this the order of perturbation theory, we do not need to worry about orientifold planes: the Möbius strip diagram cannot contribute.
2.3 Orbifold spectrum from D-branes at singularities and Yukawa couplings
We review type IIB string constructions based on orbifolds with D3/D7 branes. The orbifold point group acts on the internal space as
for complex coordinates of the three ’s in , and the corresponding worldsheet fermions . (For more details on our conventions, see appendix A.) The twist vector satisfies mod . For the spectrum to exhibit the field content of supersymmetry we require that all .
The geometrical moduli of each two-torus are
where is the angle that the complex structure forms with the -axis, and and are the radii of the two one-cycles of the torus, cf. figure 4.
We note that these parameters are not exactly the supergravity moduli fields, see Lust:2004cx (); Berg:2005ja (); Blumenhagen:2006ci () for a detailed discussion of this, but they are the ones that directly arise in our string theory calculation. In particular, is not the the imaginary part of a Kähler modulus corresponding to a four-cycle volume in the D3-D7 duality frame, as it sets a two-cycle volume, which is the imaginary part of a Kähler modulus in the D5-D9 duality frame. Also, for some orbifolds, the complex structure is actually fixed — a table can be found in Dixon:1990pc ()
We will mostly focus on compactifications based on the toroidal orbifolds , , and , as summarised in table 2. For these models, the six-dimensional compact space can be factorised into three ’s on which the individual orbifold twists act. One can classify the action of the orbifold twists on strings into three different sectors:
Completely twisted. the orbifold action on all subtori is non-trivial: all . The relevant physical fields exhibit supersymmetry.
Partially twisted: the orbifold action leaves one invariant, which we choose to be the third one, described by the coordinates and . Thus mod and mod . These sectors appear in , and orbifolds. The relevant physical fields exhibit supersymmetry.
Untwisted: the orbifold action leaves all tori invariant. These can be identified as sectors for D3-D3 states and as sectors for D3-D7 states.
A first useful step in characterising the relevant aspects of the spectrum of the global (compact) orbifold is to “zoom in” on D3-branes at local (noncompact) orbifold singularities, to establish which combinations of chiral superfields give gauge-invariant Yukawa operators. Details on this can be found in 0005067 (); Douglas:1996sw (); Douglas:1997de (). The orbifold action 8 on the string coordinates is accompanied by a transformation that acts on the Chan-Paton (CP) indices of open string states. The orbifold spectrum is found by classifying states under this combined action into invariant and non-invariant states. The result is that the gauge group is and chiral multiplets transform as bifundamentals. For an orbifold twist we have
We label chiral multiplets as where is a label for the three two-tori of the compact space and the subscripts denote the gauge groups under which the field transforms.
An allowed Yukawa coupling corresponds to a gauge-invariant combination of three such chiral superfields of the form
although an allowed coupling may not actually be present. We summarise in table 2 the combinations that rise to acceptable Yukawa couplings for various orbifold singularities used in model building.
|orbifold singularity||orbifold twist||allowed Yukawa couplings|
These expressions will be useful for the CFT calculations that follow: the labels determine the H-charges of the vertex operators to be inserted (see section A). A trivial observation is that the combination is always gauge-invariant and thus an allowed Yukawa coupling for all orbifolds.
3 De-sequestering in string perturbation theory
In this section we calculate the dependence of a D7 gauge coupling on chiral matter on D3 branes. This is the Lagrangian term
As stated in 7, the superspace expansion gives two ways to probe the existence of this term, by looking either for couplings
or for couplings
3.1 Setting up the string calculation
As we are interested in a double-trace operator, the relevant one-loop string topology is the cylinder. As shown in figure 6, the gauge boson vertex operators are inserted on one boundary of the cylinder while the scalar operators will be located on the other boundary. In appendix A we review the basic ingredients of conformal field theory that are necessary to perform the computation.
We first compute this correlator for a Yukawa operator with tree-level flavour structure , before generalising this to other flavour combinations. We begin by writing down the relevant vertex operators which we choose to be in the zero-picture for both the gauge bosons and the scalars. This will satisfy the condition that a cylinder worldsheet requires vanishing background ghost charge. The correlator is then
with the zero-picture vertex operators
We have suppressed constant polarisations, factors of the string coupling and also Chan-Paton factors, that we will restore in section 3.6.1. Here, and are complex external coordinates, and denote complex internal directions for . Note that this particular assignment for the scalars is consistent with a Yukawa coupling . We have chosen the gauge bosons to be polarised in the complex plane only (the , directions in terms of real fields), thereby breaking the symmetry between and , in fact we will often break Lorentz invariance in intermediate steps in this paper and then assure its presence at the end of the calculation. The bars on the vertex operators for spacetime scalars, are conventional; our spacetime scalars are defined with negative orbifold charge, as explained in appendix C.1. Also, the momenta are external and complexified, as in eq. 113 in the appendix.
We now make an observation that simplifies the calculation considerably: as the operators above only contain barred and fields for the internal directions, and no combination barred-unbarred occurs, the (quantum) correlators all vanish, as explained in appendix A.3.2. The worldsheet fermion pieces of the spacetime scalar vertex operators then cannot contribute to the result at all, but from the scalars there will be classical contributions. The relevant terms of the scalar vertex operators are then:
More on classical solutions is given in A.3.2.
In all, the vertex operator for a scalar from the chiral multiplet will just contain bosonic fields including . We continue the calculation for the case of the Yukawa coupling and then re-evaluate our results for the other cases.
In intermediate results, we will suppress overall factors and factors of .
3.2 The cylinder correlation function
At this point our correlation function 15 is given by:
The first factor is essentially a well-known one-loop two-point function of gauge bosons (see e.g. Kiritsis:1997hj (); Antoniadis:2002cs (); Berg:2004sj ()). For completeness we sketch this calculation here. It splits into four terms, but the two cross terms clearly vanish. The term actually vanishes, as we proceed to show. It consists only of the bosonic fields:
Contraction of all terms gives
where is some function of the insertion points and the sum is over all spin structures . Bosons are ignorant of the spin structure, so we were able to pull out of the sum, then all that remains is the partition function which vanishes in a supersymmetric model. The amplitude therefore reduces to the fourth term in 3.2:
which splits into correlation functions involving the fermions, the momentum exponentials and the internal bosonic fields. As the fermion correlator is universal to all cases we evaluate it next.
3.3 Fermion part of gauge-boson two-point-function
The factor encodes the kinematics of the gauge-boson kinetic term: , which we can formally extract, so in the rest of the calculation we suppress this factor. To sum over spin structures, we reexpress Kiritsis:1997hj (); Antoniadis:2002cs (); 0607224 ()
where , see e.g. 0607224 (). Here is the Weierstrass function
For comments on these relations, see appendix A.4.
As the Weierstrass -function and the piece of are both independent of spin structure , the summation over spin structures for these terms produces the supersymmetric partition function, which vanishes. Therefore this correlator only receives contributions from
Now as we can write the contributions of the fermionic parts of gauge bosons to the overall amplitude as
where primes denote derivatives with respect to . We see that the contribution is independent of the worldsheet positions of the gauge bosons, a general phenomenon in orbifold sectors with enhanced supersymmetry. Later, we will combine this result with the bosonic correlators and the fermionic partition function. Next, we consider another ingredient of the calculation — the classical solutions over winding modes.
3.4 Correlator of internal bosonic fields — sums over winding modes
The part of the overall correlation function that is sensitive to the flavour structure of the Yukawa coupling is the correlator of internal bosonic fields. A potential Yukawa coupling of the form is probed by a correlator
As we discuss in detail in appendix A.3.2, this has a classical part which is found as a sum over classical solutions
Non-trivial classical backgrounds correspond to winding modes of stretched strings, so we need to examine the appearance of such modes in our setup. We will use the classification in section 2.3 to distinguish the various actions of the orbifold on the (super-)string coordinates.
We begin by examining the case where both gauge bosons and chiral matter are located on stacks of D3-branes. We generate chiral matter by placing a D3 brane at an orbifold singularity at , and place another stack supporting the gauge bosons away from the origin at where we choose the two stacks of branes only to be separated in the third 2-torus by a distance . (For our convention about this, see A.3.2.) This setup is general enough to examine the desired physical effects.
Winding solutions for open strings exist for directions with Dirichlet-Dirichlet (DD) boundary conditions. In principle, strings can stretch in any of compact dimensions for this setup. However the contribution of such modes depends on the orbifold twists (see section 2.3):
Completely twisted directions do not allow for winding modes as they are projected out. A classical solution cannot be given for any combination of and thus completely twisted sectors do not contribute to the overall result.
In partially twisted sectors, winding is only allowed along directions on the untwisted 2-torus, which we have chosen to be the third, and only combinations of fields polarised in this 2-torus can contribute. Thus only three-scalar-operators of the form give a non-vanishing classical solution whereas other configurations do not contribute.
Untwisted sectors can in principle support winding modes on all of the subtori. However, this is not actually relevant as the contributions of the untwisted sectors are zero due to the vanishing of the quantum correlator for supersymmetry. This will be different for D3-D7 models.
To summarise, for D3-D3 models the only contribution to the amplitude arises from Yukawa couplings of the form where labels a 2-torus that is left untwisted in a orbifold sector. Gauge-invariant operators of this form only arise in the following orbifolds: with and with . The relevant superfield term there is and the classical solution is then given by (see A.3.2):
where we defined
and we have from 128 that
and where is the partition function over winding modes, as given in equation (139).
In this case the gauge fields are located on a stack of D7-branes whereas chiral superfields are supported on a stack of D3-branes. We let the D7-branes wrap the first two subtori and be located at of the third 2-torus, with D3-branes at the origin. We first note that D3-D7 winding modes are only allowed on the third torus, as on the first two subtori the Neumann-Dirichlet boundary conditions prohibit such modes. Thus the classical solution automatically vanishes for any combination of fields except .
As before, winding states are only allowed on untwisted tori. We therefore restricted to partially twisted and untwisted sectors of the orbifold as defined above. For D3/D7 systems the untwisted sectors only preserve supersymmetry, and so the correlator does not vanish and such sectors can give non-zero results. This gives an effect in more models than for the D3-D3 case where only and models were interesting — here we also have non-zero correlators for in the untwisted sector of orbifolds.
In an orbifold we also need to include images of the D7-branes if they are not at orbifold fixed points. For each stack of D7-branes we add identical stacks at the image loci of the original stack. Thus, for D7-branes located at on the third 2-torus we include images at for . To obtain a result invariant under the orbifold, we also include strings stretching between the D3-branes and all the image D7-branes. In our present calculation this leads to the following modification: each stack of D7-branes will give rise to winding solutions stretching from the D3-stack to the D7-stack. The corresponding classical partition function is the same for strings stretching between the D3-branes and any of the stacks of D7-branes. This is due to the fact that the lattice used to construct the torus is invariant under the orbifold action. However, as for image branes the internal correlator is modified accordingly:
for . However, the only orbifold models that allow for a non-vanishing correlator are generated by the point groups , or for which (see table 2). Correspondingly, all images contribute the same result:
3.5 Completing the calculation
In this section we combine the results of the previous sections. We also need to include the correlator over momentum exponentials which contributes
where is the Green’s function with NN boundary conditions. The amplitude is then integrated over worldsheet positions and the modular parameter of the cylinder .
In §3.4 we established that the only non-zero contribution arises in partially twisted sectors of the orbifold. To be specific we take mod and mod . Collating the previous results, the complete amplitude is
where is the partition function given in equation (139). We can apply the Riemann identity (169) to collapse the sum over spin structures to a constant.666If we repeated the analysis for the untwisted orbifold sector, the sum over spin-structures would be identically zero due to (167). Hence untwisted sectors do not contribute in the D3-D3 case. Further, in the partially twisted sectors of the relevant orbifolds ( sector of both and ) we have . Thus we are left with:
Before analysing this expression we present the result for the D3-D7 model as it will be identical.
For the case of D3-D7 models the untwisted sector only preserves (in contrast to for D3-D3) and we thus expect a non-zero result. We combine our previous results with the appropriate partition function to obtain
with again given by (139). As above we get
which is identical to (38) and hence we discuss both results together.
3.6 Result — evaluating the integral over
We now perform the integrals in equations (38) and (40). The only momentum-dependent term is the correlator over momentum exponentials. This is also the only term depending on the worldsheet positions. As we are only interested in the amplitude at vanishing momenta , the integrand can be considerably simplified. After the spin structure sum has collapsed to a number, there is no possible source of pinching singularities that could make the exponential of bosonic propagators contribute. The limit can then be taken without subtleties, removing any dependence of the amplitude on world-sheet positions. Subsequently, the integral over world-sheet positions can be performed giving a contribution .
Then, ignoring other overall factors, we want to compute the following integral over the world-sheet modulus , where we now include the winding modes explicitly:
The double sum over and is of the form
where we identified the first derivative of the Weierstrass function. More explicitly, our first answer for the amplitude is
We can arrive at the result (44) from 41 by a different method. In the integrand, we restore the triple derivative from above and instead of 41, we recast the sum over winding modes as a generalised (Riemann) theta function (163):
where is the metric on the torus wrapped (92), is a real 2-component vector with and is a numerical constant which we leave undetermined. We now want to move the derivatives with respect to the complex separation outside of the integral over . Unlike in the previous method, the integral now apparently becomes divergent, so we introduce a UV cutoff in this intermediate step. Fortunately, we know from the previous method that the result is finite, so this is no cause for concern. The remaining integral is now the same as that which occurs in usual gauge coupling renormalisation (i.e. without the three scalar insertions we have here), so we can perform the integral as in 0404087 () to find:
with the scalar boson propagator on the torus777see e.g. section 7.2 of Polchinski1 (). Note that we give two arguments for , because although here the transverse space is flat and we have translational invariance, in general one might be interested in background gravitational and -form fields in which case the Green’s function is not a function only of the difference .
i.e. is the solution of the Laplace equation on the torus with a delta function source and a neutralising background charge. Happily, as expected from our previous calculation, the cutoff disappears from the final result in virtue of the differentiation. Further, we see that the last term of the above expression is proportional to and will be annihilated by the three derivatives. We find 888The sum and product expressions for the theta functions only converge for in this case, which is unconventional. To recover the convention, we could switch the convention of negative orbifold charge for the scalars, so those vertex operators become unbarred, and simultaneously switch the definition of to .
We have performed the integral over the cylinder worldsheet modulus in two different ways, and we would like to highlight the differences between the two calculations. If there had been a lower power of than two or a lower power of than three in 41 in 42, the integral would have failed to converge, just like the integral did in the second method when we peeled off the three derivatives. These powers originally came from from three holomorphic vertex operators for the three spacetime scalars in the Yukawa-like coupling, and this is an example of the good behaviour of this amplitude alluded to in section 2.2. In the analogous calculation of the gauge coupling correction in Berg:2004sj (), without the Yukawa-style triple scalar insertions, those divergences appeared in each separate diagram due to tadpoles in the open-string UV limit, and could only be cancelled between diagrams by enforcing tadpole cancellation. We see that for the particular calculation of interest in this paper, this is not a problem.
The above results give the dependence of the gauge kinetic function on a stack of D3/D7-branes due to chiral matter on another stack of D3-branes. As discussed in section 12, the term in the superspace action we are interested in is
First, we will see if we can understand the scaling behaviour obtained in 48. For this purpose, let us set the two torus radii equal: and . We find
We can in fact understand the scaling from an analysis in four-dimensional effective supergravity. We have computed the superfield component coupling999Note that this is Weyl invariant, so it is not affected by the Weyl rescaling we need to go to Einstein frame.
in the string effective action. In terms of superfields, we are looking at a term in the gauge kinetic function . The interaction 51 comes from the term 49 in the superspace action, once the matter fields are canonically normalised. The Kähler metric for matter on D3 branes scales as . Performing the canonical normalisation and restricting to flat space, we obtain
where . The scaling of (50) is then consistent with supergravity expectations.
Our second comment is that as noted in the introduction, 49 generates two distinct Lagrangian terms,