Aspects of Low Energy Effective String Theory

1 Introduction 3

1.1 Motivations . . . 3

1.2 Outline . . . 4

2 Effective field theories 6 2.1 Non linear symmetry . . . 6

2.2 The coupling H†HF2 . . . 7

2.2.1 Structure of the coupling . . . 7

2.2.2 Minimal coupling . . . 9

3 String Theory 12 3.1 Open strings and Branes . . . 12

3.1.1 Boundary conditions . . . 12

3.1.2 Intersecting branes . . . 14

3.1.3 Superstring theories in ten dimensions . . . 16

3.2 Low energy effective theories . . . 17

3.2.1 Compactification on a six Torus . . . 17

3.2.2 The model . . . 18

3.2.3 Branes and supersymmetries . . . 21

3.3 Scattering Amplitudes . . . 22

3.3.1 Topology of world-sheets . . . 22

3.3.2 Insertion of VO’s . . . 24

3.3.3 Conformal Killing vectors and Pictures . . . 25

3.4 Conformal Field Theory . . . 27

3.4.1 Disk . . . 27

3.4.2 Torus and Annulus . . . 27

4 Four points amplitude 30 4.1 External Magnetic Field . . . 30

4.1.1 Open string in background magnetic field . . . 30

4.1.2 Spectrum of open strings at the intersection . . . 32

4.2 Vertex Operators . . . 33

4.2.2 Higgs Vertex Operator . . . 34

4.3 Tree Level Amplitude . . . 37

4.4 Three Intersecting Stacks . . . 38

4.5 N=1 Amplitude . . . 39

4.5.1 Neutral Higgs . . . 40

4.5.2 Charged Higgs . . . 46

4.6 Non Supersymmetric case . . . 48

5 Conclusions 50 5.1 Results . . . 50 5.2 Future developments . . . 52 A Notation 53 B Green functions 56 C Theta function 58 D QFT diagrams 60

Introduction

1.1

Motivations

One of the most important issues of modern theoretical physics is the under-standing dynamic of electroweak symmetry braking and Higgs mechanism. Mass generation of gauge vectors responsible of weak interaction has kept the attention of physicist for entire decades.

Important result have been achieved towards years, starting from Higgs phe-nomenon until Glashow-Salam-Weinberg formulation of Standard Model. The goal of complete explanation of such mechanism, however, is far from being reached.

Many other hypothesis have been considered, most of which have been com-pletely or partially ruled out from measurement results coming from LEP2. A typical example is represented by supersymmtry: introduced to stabilize loop cor-rections to Higgs mass has lost some of its original justifications, since absence of light Higgs forces supersymmetric theories to be unnaturally fine-tuned.

With incoming of LHC, a new range of energies will be accessible: precision tests will help to discern, among all developed models, those that better describes fundamental interactions.

In present context it becomes fundamental being able to predict what kind of scenarios could verify if a particular description is used instead of another: lots of recent works are dedicated to analyze possible results of precision test.

Moreover in recent years many interesting developments have been achieved in theoretical physics and have conduced to completely innovative formulations of symmetry breaking mechanism. One example is represented by compactification of extra-dimension, where higgs mechanism is implemented by self interacting potential coming from dimensional reduction. Other theories allow the possibility that symmetry breaking is implemented by new sectors strongly interacting at hight scale ([1]).

All these models condivide the general belief that dynamics investigated until present days are just a low energy manifestation of a more complex structure that will be partially revealed by experiments at Tev scale.

Moreover all theories formulated are based on the fundamental hypothesis of minimal coupling, namely that at leading approximation (tree level) the only coupling between matter and gauge vector fields is introduced in the effective lagrangian by covariant derivatives.

Present work wants to investigate this hypothesis in the framework of effective lagrangian describing low energy string theory. Superstrings theory remains one of the best motivated and most encompassing approach to extending the Standard Model and answering questions the latter does not address. Several promising models conducing to MSSM have been developed ([3]), even if not in full details.

String theory represents our starting point: since it can be formulated in a self-containing way, the complete theory is known and we are able, at least in simplified models, to write down exact effective lagrangian, integrating out all unobservable states.

1.2

Outline

In order to perform concrete calculations we consider a simplified model of type I superstrings. Four dimensional theory is obtained compactifing extra dimensions on a six-torus. We than simulate Higgs field taking an open string stretching between two intersecting stacks of branes. This allows to attach a gauge group U (2)× U(1) to the string and, more important, to break down supersymmetry from a starting configuration of N=4. We chose to preserve however, at least at tree level, N=2 supersymmetries.

A string stretching between intersecting branes satisfies unusual boundary conditions: oscillators carrying internal indexes1 have their commutation relation changed and can originate light or massless states. Higgs fields is indeed the mass-less scalars coming from the mechanism just discussed.

We investigate how Higgs field is coupled to gauge vectors in the effective low energy lagrangian. To this aim we integrate out all heavy state of the theory. This is easily obtained since string theory scattering amplitudes sum automatically all possible contributions to one process.

The unique gauge invariant structure that could describe first order correction due to processes mediated by heavier states is the six dimensional operator:

H†HFµνFµν

where H is Higgs field while F is the gauge field strength. Its coefficient is evaluated computing scattering amplitude of two Higgses and two gauge vectors at tree level first and at one loop afterwards. We analyze both the case in which H is charged or neutral under the gauge group of the field strength Fµν.

Present work is organized as follows: chapter two is dedicated to a review of general structure of effective theories with spontaneous symmetry breaking.

1_{Remind that in a compactified theory fields that in original theory carry a vector index separe}

Then we analyze the contribution of the operator H†HF2 in a scattering process involving two Higgses and two vectors from a QFT point of view. In this way we can easily distinguish interesting terms in string amplitude. Finally a minimally coupled theory is taken in account and prediction concerning results are given.

Third chapter contains a summary of results and notions of string theory that are used later. Moreover the model we use is presented and discussed in details.

Forth chapter contains the original part of the work. Tree level approximation is computed evaluating disk amplitude or looking at open string spectrum modi-fications in presence of a background gauge field. Than, one loop corrections are calculated towards the study of annulus amplitude.

Finally all results and conclusions are collected in last chapter.

Acknowledgements

We would like to thank professor Riccardo Rattazzi for having contributed to raising the question and for the helpful discussions on effective field theories.

Effective field theories

2.1

Non linear symmetry

In what follows we deal with a general framework that can be applied to many different contexts describing effective theories. We start with a generic field theory on which is realized a representation of a group G, whose elements are called g. We suppose that fields contained in the theory can be divided in those forming low energy spectrum and heavier states, characterized by a energy scale Ms ([1]). We assume than that by some unknown mechanism the groupG is broken down to a subgroupH whose elements will be denoted h. The parameter responsible of the breaking will be called gB. We take G to be a compact Lie group: the breaking of such a group involve the presence of a Goldstone boson H.

The theory is also coupled to gauge fields through a set of constants generally denoted with gSM. We assume that, in the limit gSM −→ 0, H is a massless Goldstone mode, while gauging to vectors will produce mass corrections.

We want to investigate the low energy effective lagrangian for energies much lower than Ms, obtained integrating out all heavy fields. In particular we are interested in effective description of interaction among Goldstone bosons and gauge vectors.

It can be shown that the broken mechanism together with couplings to all light fields can be parameterized by a serie of increasing dimension operator. The resulting lagrangian is invariant under particular non linear transformations of fields that become linear when restricted to subgroup H. In the two works of Coleman & al. ([4],[5]) all non linear representations are classified and the general method to construct invariant Lagrangian is given.

Here we generally write effective lagrangian in the form of an expansion in powers of derivatives of the Goldstone field, suppressed by an appropriate power of a mass that describes the scale of symmetry breaking:

L = L(4)+ 1 M_s2L

(6)_{+ O(M}−4

Moreover each of these terms allows a loop expansion weighted by an increasing power of parameter gB or gY M. Notice that we have chosen the mass responsible of symmetry breaking to be Ms. We have in mind the behavior of a string theory compactified on a six torus with radius:

RT ∼ 1 Mplank and Ms∼

√

α
is the scale of heavier state mass. In this case gB can be chosen to be:

gstring = g2_B (2.2)

2.2

The coupling

H

HF

Second term of effective lagrangian expansion (2.1) contains higher derivative cor-rections to Higgs couplings. Since Lagrangian must be gauge invariant we can easily select terms that describe interactions between two gauge vectors and two photons.

2.2.1 Structure of the coupling

Let’s consider a process with two photons in the initial state with momenta k₁, k₂ and polarizations ξ₁, ξ₂ and two Higgses in the final state with with momenta −p1,−p2. The matrix element between these states is:

Sf i= ξµ₁ξ₂ν < 0|h(−p₁)˜h(−p₂)
i  Lint_(x)d4_x  a†_µ(k₁) a†_ν(k₂)|0 > (2.3) where, for the moment, we consider the interaction mediated only by the operator: Lint_{(x) = ge}iP₀t_HH†_F2_{(0, x)e}iP₀t _(2.4) Substituting in previous expression gives:

Sf i= i(2π)δ(Ef−Ei)gξ₁µξ₂ν < 0|h(−p1)˜h(−p2)
 HH†F2(0, x)d3x  a†_µ(k₁) a†_ν(k₂)|0 > (2.5) Since the fields in the matrix element are free, they can be expressed by usual modes expansion: H(x) =  d3q 2q₀(2π)3 

h(q)e−iqx+ ˜h†(q)eiqx 

H†(x) =  d3q 2q₀(2π)3  ˜

h(q)e−iqx+ H†(q)eiqx  Aµ(x) =  d3q 2q₀(2π)3 

aµ(q)e−iqx+ a†µ(q)eiqx 

(2.6) Well known expression for the field strength reads:

FµνFµν = (∂µAν− ∂νAµ)2 = 2(∂µAν)(∂µAν)− 2(∂µAν)(∂νAµ) (2.7) In the end, term between brackets becomes:

Sf i=−i2g(2π)4δ4(p₁+ p₂+ k₁+ k₂)ξ₁µξ₂ν{2(k₁· k₂)ηµν− 2k1νk2µ} (2.8) It is important distinguishing contributions coming from coupling we are interested in from other ones. The only other coupling gauge invariant that can originate terms with two photons, two Higgses and two momenta could be:

DµH†DνHFµν −→ −iq(∂µH†)AνH(∂µAν−∂νAµ)+iq(∂νH)AµH†(∂µAν−∂νAµ) (2.9) However this term is present only if the Higgs is charged under the gauge group1. The matrix elements has the following form:

Sf i
= −q(2π)4δ4(p1+ p2+ k1+ k2)ξ µ 1ξν2  (p₂+ p₁)ρ(k1+ k2)ρηµν− − (p2+ p₁)µk_2ν− (p₁+ p₂)νk2µ  (2.10) On shall : p₁+ p₂+ k₁+ k₂ = 0 p2_j = k_i2 = 0 ξj· kj = 0 (2.11) we have: Sf i
= −q(2π)4δ4(p1+ p2+ k1+ k2)ξ µ 1ξ2ν  2(k₁· k₂)ηµν − 2k2µk1ν  (2.12) Evaluating the amplitude on shall we have discovered that the structure of this term is equal to the previous one. This is right since we can integrate one covariant derivative by parts and obtain:

1_{We will see in next chapters that in string theory this configuration is completely different}

DµH†DνH Fµν =−H†DµDνH Fµν− H†DνH ∂µFµν (2.13) Anti-symmetrizing and using [Dµ,Dν] =−iqFµν we have:

DµH†DνH Fµν ₌ iq 2H

†_{H F}2_{− H}†_{DνH ∂µF}µν _(2.14) However the second terms doesn’t contribute in a scattering with two Higgses and two photons. Than we are left with a total structure:

g₁+iq

2g2 

H†H F2∝ (k₁· k₂)(ξ₁· ξ₂)− (k₁· ξ₂)(k₂· ξ₁) (2.15) Finally we can think about terms of the form:

µνρσDµH†DνHFρσ H†HµνρσFµνFρσ (2.16) However they can be eliminated as well: the first one can be reconducted to HH†F2 towards an integration by parts, while the second is null if H =const, so it brings an additional momenta and doesn’t appare in the amplitude we are considering. Non abelian case is a little bite more subtle but the structure is similar.

Let’s consider now the generic contribution given by total effective lagrangian to the scattering: < hh|Lint|AA >∝
f(4)(s, u, t, gSM) + 1 M_s2f (6)_{(s, u, t, g} SM) + . . .  ξ₁µξ₂ν(k₁k₂ηµν−k_2µk_1ν) (2.17)

In our particular case f(6) is constant in external momenta. Notice that the cou-pling H†HF2can be regarded as a mass term for Higgs field. Therefore its presence should be originated by a breaking of residual Goldstone symmetry.

2.2.2 Minimal coupling

Consider the general theory introduced at the beginning of the chapter. Low energy interactions are determined by usual Feynman diagrams deriving from kinetic terms and vertices for light fields, plus a bunch of higher dimensional operators describing the action of having integrated out heavy states.

Let’s consider again a scattering process involving tho Higgses and two photons or two gluons: in the hypothesis of minimal coupling there is no ordinary diagrams contributing to the amplitude. Indeed, since Higgs is neutral under electromagnetic field, covariant derivative acting on the scalar boson does not contains any photon field. We still have to take in account processes mediated by heavy states: however, since no such state is present between incoming or outcoming particle and we do not consider loop diagrams, only possible contribution are those show in figure.

Figure 2.1: Feynman diagrams for scalar QED. All vertices come from minimal coupling in covariant derivatives.

Whatever is the spectrum of the theory considered and coupling among different fields, vertices present in (2.2) are not in accord with minimal coupling.

Similar it is the case of Higgs charged under the gauge group of which vectors belong to. There will be some additional diagrams coming from covariant deriv-atives but the contribution to the amplitude but they have a different kinematic structure (see figure(2.1)).

We can rule out the birth at tree level of the operator H†HF2 in a minimally coupled theory.

Figure 2.2: Possibile tree level diagrams for Higgs neutral under vectors gauge group. Dashed lines represents hypothetical heavy intermediate states.

Let’s move to next to leading approximation and consider one loop contribu-tions to amplitude. We first discuss the case of neutral Higgs. The only diagram in low energy effective field theory is represented by a process mediated by a loop of bosons or fermions connected to Higgs couple by a gauge vector propagator (first picture of figure (2.3)). Fields that create the loop should be charged under final gauge vectors and under a second group common to Higgs as well2. In the appendix we show that with following choice for polarizations and momenta3:

kµ₁ = (k, k, 0, 0) kµ₂ = (k, 0, k, 0) (2.18)

2_{In supersymmetric standard model the loop can be originated by quarks and bosonic}

super-parters

ξi· kj = ξi· pj = 0 (2.19) contribution of all this diagrams reduces to :

AQF T ∝ u− t

s (2.20)

Finally we consider a charged Higgs. Number of diagrams grows rapidly, es-pecially for non-abelian case. Let’s focus for example in a U(1) gauge group. In addition to previous process we can construct events such those shown in figure (2.3).

Figure 2.3:Left: Only possible one loop diagrams in a minimal coupled QFT using low energy fields and neutral Higgs. Center and Right: diagrams emerging for charged Higgs case.

Previous diagrams are QFT contributions to the amplitude: they come from interactions between fields present in low energy effective lagrangian. The string amplitude that we compute in chapter four contains latter terms and all other processes mediated by heavier states. Operator H†HF2 represents at first order all contributions of heavy fieds. In order to extract its coefficient we must subtract QFT terms.

At one string loop approximation there is no reason to expect coefficient of H†HF2 to be zero.

String Theory

This chapter represents a brief introduction to topics that are usually not described in much details in academic texts. We want to justify some formula and discuss some theoretical and technical aspects of String Theory, with particular attention to the framework in which we have performed our computations.

Section 3.1 deals with the concept of Dp-brane and states at the intersections. In the following section we describe how a four dimensional theory can be obtained starting from a ten dimensional one. Our model is than introduced. Section 3.3 is dedicated to scattering of strings while in last section some technical aspects concerning the calculation we are going to present are discussed.

3.1

Open strings and Branes

3.1.1 Boundary conditions

Bosonic string action reads:

S =− 1 4πα



d2σ∂aXµ∂aXµ (3.1)

where we have exploited conformal symmetry and diffeomorphism symmetry to put two dimensional metric γab in lorentzian form ηab. Variation respect to Xν gives free wave equations plus surface terms that must vanish. This can be achieved imposing suitable boundary conditions (BC). Open string BC read:

na∂σXν(τ, σ) = 0 σ ∈ ∂M, na⊥ ∂M Neumann BC (3.2) while for closed strings BC used are:

Xν(τ, σ) = Xν(τ, σ + π) Periodic BC (3.3)

Moreover we define Dirichlet BC:

These unusual conditions will emerge in the following, in the dual description of open strings.

Bosonic string theory has a duplex interpretation: it’s a two dimensional con-formal field theory (CFT) with an internal global symmetry group SO(1, d− 1) but the quantity Xµ in (3.1) is commonly regarded as an embedding of a two dimensional manifold called world-sheet in d-dimensional flat target space.

We want now to analyze the case in which one coordinate is compactified on a circle:

Xd−1  Xd−1+ 2πR (3.5)

A first effect of this periodicity is to quantize the momentum1 of the string in the d− 1 direction. Indeed e2πRp(d−1)_{, the translation operator along X}d−1_{, must be} periodic; than:

pd−1 = n

R n∈ Z (3.6)

In closed string theory there is a second effect, namely the possibility that a string winds around compactified coordinate, but this is absent in open string case, since it can always be unwound from periodic dimension [14].

Mass formula in present case reads:

α
M2= Nd−1− 1 + n

2

R2 (3.7)

where Nd−1 is the operator that counts number of excitations in d−1 dimensions2. We can notice that in the limit of small radius the momentum pd−1 is frozen to n = 0 and the string lives in d-1 dimensions only3. However this theory admits a dual description in terms of a new variable; if we call XR(L) the right(left) moving solution of free wave equation, we have:

Xd−1= X_Ld−1+ X_Rd−1 ←→ X
d−1 = X_Ld−1− X_Rd−1 (3.8) We obtain a new open strings theory with the same energy momentum tensor but Neumann BC of original coordinate translate in Dirichlet BC for dual one:

∂σXd−1=−i∂τX
_d−1

(3.9) Thus both ends of an open strings are fixed in time. Moreover if we compute:

X
d−1(π)− X
d−1(0) =  π 0 ∂σX
_d−1 =−2πα
pd−1=−2πn α
R (3.10)

1_{The momentum is defined to be the conjugate variable of X in Hamiltonian formalism or}

more simply the coefficient of the term linear in τ in X(τ, σ) expression

2_{All precise expressions are given in the appendix}

3_{This is the same behavior of QFT; on contrary closed string keep on living in d dimension}

that is, X
d−1 is still a periodic function provide we identify α_R
with new com-pactification radius R
. Than, in dual picture, the limit R −→ 0 corrisponds to a d-dimensional theory in which ends of open strings are fixed on a hyperplane called D(d-2)-brane.

Until now we have considered only oriented strings; we now want to introduce the concept of unoriented string. Let’s define on the world-sheet the symmetry transformation:

Ω : σ→ π − σ (3.11)

We impose to the states of the theory to be invariant4 under Ω. Its easy to see that latter operator acts on string oscillators as:

Ωαi_nΩ−1 = (−1)nαi_n (3.12)

in open string case while exchanges left with right moving modes for closed strings. Many states are thus ruled out from the spectrum ([14], [15]).

Let’s than compactify a closed unoriented string on a cirle and consider T-dual description. In original theory symmetry transformations acts as:

Ω : XR ↔ XL (3.13)

thus dual coordinate is transformed as:

Ω : XL(τ + σ)− XR(τ − σ) ↔ XL(τ − σ) − XR(τ + σ) (3.14) For compact coordinates Ω acts as a spacetime parity plus a world-sheet par-ity. Projection on eigenstates doesn’t restrict the spectrum5 but simply relates different points of the string: the effective compact space in not more a circle but:

S1 −→ S1/Z₂ (3.15)

with two fixed hyperplanes at extremities called orientifolds. For open strings it is similar a part the presence of branes.

3.1.2 Intersecting branes

We can keep on compactifing and dualizing in more than one coordinate and create Dp-branes that extend only in few directions6. In this way ends of open strings are constrained in lower dimensional hyperplanes. It’s also possible separate and rotate these objects. Strings stretching among such branes can originate massive states at energies much lower than string scale and even a chiral spectrum ([2]).

For the particular work we are presenting we do not need to create parallel stacks of branes so we do not focus on this topic. More useful is instead the

4_{eigenvectors to eigenvalue +1.}

5_{In spacetime the projection does restrict the spectrum but not in compact space.} 6_{Dp-branes are p+1-dimensional objects extending in p spatial dimensions plus time}

mechanism that permits us to built intersecting branes at generic angles. To do this we switch on a constant magnetic field in compactified space; in section 4.1 we will follow a similar procedure, however we should notice that now we are working in dual representation, while in next chapter the magnetic field is in spacetime and we do not dualize any coordinate.

Let’s suppose that a stacks of brane extends in 1,2 directions. We choose A₁= 0, A₂ = F X₁, so that F₁₂= F .

The coupling to a magnetic field for a bosonic string is obtained adding to (3.1) the term: qLF  dτ X₁∂τX2   σ=0 + qRF  dτ X₁∂τX2   σ=π (3.16) New boundary conditions for dual coordinate read:

∂σ(cos βLX₁
+ sin βLX₂
) = 0

∂τ(− sin βLX₁
+ cos βLX₂
) = 0 σ = 0 (3.17)

∂σ(cos βRX₁
+ sin βRX₂
) = 0

∂τ(− sin βRX₁
+ cos βRX₂
) = 0 σ = π (3.18) where we have defined tan βL = 2πα
qLF and tan βR = −2πα
qRF . These are boundary conditions for a string whose ends are confined on 2 stacks of branes that make, respectively, an angle βL and βR with horizontal axis. It is clear that only the difference of angles, and than sum of charges, is important.

To summarize, we start with a collection of coincident branes, whose number determines the gauge groups U(N) of Chan-Paton factor. Adding magnetic fields in directions parallel to branes permits us to rotate some of them, according to charges qjL+ qjRwhere j = 1, . . . , N labels not necessarily different charges.

We now want to describe the spectrum of a bosonic string stretching among two intersecting branes. If we define Z = X₁+ iX₂ we can rewrite previous BC as: Re ∂σ  e−iβL_Z  = 0 Im ∂τ  e−iβL_Z  = 0 σ = 0 (3.19) Re ∂σ  e−iβR_Z  = 0 Im ∂τ  e−iβR_Z  = 0 σ = π (3.20)

Z(τ, σ) = z + ieiβL α
2 n∈Z
zn_+ n + e −i(n+)(τ−σ)−₋ z−n− n + e i(n+)(τ+σ)  (3.21)

and the following canonic commutation relations hold:

[zm−, zn+] = (m− )δm n (3.22)

We finally have the mass formula: α
M2 = Nd−2+ ∞ n=−∞ : z_−n−zn+ :− d− 2 24 + 1 2(1− ) (3.23) Here we have made the hypothesis that only two coordinate are compactified and the other d− 2 satisfy standard boundary conditions, otherwise we would have a factor 1₂(1− ) − ₂₄2 for each complex fermions7 like Z. Note that oscillator that annihilate the vacuum are those with positive mode number: normal ordering : z_−n−zn+: poses on the left the one with negative frequency.

3.1.3 Superstring theories in ten dimensions

Closed superstring theories can be self consistent in ten dimension: the request for this to happen is the modular invariance, namely invariance under transformation of modular group:

SL(2, Z) (3.24)

Two different theories can be defined, both with two Majorana-Weyl supercharges, according to if latter have the same chirality (type IIA) or opposite chirality (type IIB).

On contrary, type one superstring theory cannot be constructed starting from only closed or open strings. Tadpoles emerges at one loop: in order to cancel these divergences we must sum amplitudes on cylinder, Klain bottle and Moebius strip. Thus closed unoriented and open strings must be in interaction.

It can be shown that type II theories can be obtained as a dual description of an open unoriented superstring theory. Dualizing the latter produce Dp branes and orientiforlds planes. Bulk theory is described by type IIA or IIB respectively if we have dualized an odd or even number of coordinates: in particular original theory, a type one with D9-branes and O(9) orientifolds, coincides with type IIB. Near O(p) orientifolds and Dp-branes the theory in that of an open plus closed oriented N=1 superstring.

7_{The proof that the contribution to 0-point energy of a complex boson with expansion (3.21)}

3.2

Low energy effective theories

String theory is a quantum theory of interacting fields with growing spin and mass. Its spectrum is characterized by levels spaced by an energy proportional to string scale (α
)−1/2. This means that at energies much lower than latter scale only lighter fields of spectrum can propagate, while all others are indirectly present through loops. A simple example is given by open superstring theory in flat background space: its massless spectrum gives N=1, D=10 supergravity.

As in the latter case these low energy theories can be described by an effective lagrangian that contains low energy fields and reproduces in QFT all amplitudes calculated by string formalism.

3.2.1 Compactification on a six Torus

String and superstrings theories are respectively twenty-six and ten dimensional theories. In order to deal with phenomenology we must reduce them to ordinary four dimensional world. As consequence, additional dimensions should be elimi-nated from the world volume in which fields can move. In field theory the procedure is quite standard: one can choose for example an ansatz for the D dimensional background metric of the form:

gM N =  f (y)ηµν 0 0 g(y)ij  (3.25) where we have separated spacetime variables xµ, µ = 0, . . . , 3 from internal vari-ables yi, i = 4, . . . , D. Moreover we have made the assumption of four dimensional Poincar´e invariance.

The same procedure can be brought off in strings frame. We present here a particular case in which the internal space is a six torus. This is one of the simplest compactification that can be performed but it lets us exploit general properties of string theory and make explicit calculations.

We consider than a superstring theory in which ten dimensional background is a product of four dimensional Minkowski space and factorizable six dimensional torus:

M10=R4⊗ T₁2⊗ T₂2⊗ T₃2 (3.26) In a given manifold we can define homotopy classes considering the equivalence class of all curves that can be connected each other by a continuum transformation. Clearly if the manifold has a trivial topology we have only one equivalence class and the manifold is said simply connected. A torus has Euler number χ = 0, than we can define two different equivalence classes whose elements are called 1-cycles. In general a p-cycle is a non trivial p-dimensional submanifold.

3.2.2 The model

We start with a stacks of three D9-brane and we switch on a magnetic field in compact space in order to tilt one of them8. We get a U(2) stack of D6-brane intersecting a third D6-brane9. Each brane extends in four dimensional Minkowski space plus a three cycle in compact space. We assume that 3-cycles are factorizable in 1-cycles of different tori:

c3 = c1₁⊗ c1₂⊗ c1₃ (3.27)

This means that each brane extends in one direction of each torus. If we look at three planes (X2k, X2k+1) we have lines intersecting at angles πk as showed in figure (3.1). We consider−1₂ < k< 1₂.

Figure 3.1:D6-branes extends in spacetime plus three directions in internal space, one for each two plane (X2k, X2k+1), forming angles πk.

We are interested in the low energy spectrum localized at intersection. We start considering strings with both ends on the same stack. Type I superstring theory has N=16 supersymmetry given by a Majorana-Weyl spinor in ten dimen-sions. Since we compactify on a flat manifold10 the number of supersymmetries is preserved thus we are left with a N=4 four dimensional theory. We notice the presence of vector multiplets in adjoint representation of group U(N), where N is the number of branes in the stack.

We move on spectrum of superstring with ends on different stacks. Notice that stacks intersection is a four dimensional submanifold and coincide with the space-time. Thus coordinate Xµ and ψµ satisfy free boundary condition. The solution is the standard one:

Xµ= xµ+ 2α
pµ+ i α
2 n=0
1 nx µ ne−in(τ−σ)+ 1 nx µ ne−in(τ+σ)  (3.28)

while fermionic superpartners:

8_{The theory contains also a O(9) orietifold that transforms into O(6) orientifold hyperplanes}

in dual description. However we keep the intersecting brane far from them.

9_{Remind that in order to interpret a magnetic field as a tilted brane we have to pass in dual}

picture and than one of Neumann BC becomes a Dirichlet BC

ψ₋µ = √1 2 r∈Z+a ψ_rµe−ir(τ−σ) a = 0 R 1 2 N S ψµ₊= √1 2 r∈Z+a ψµ_re−ir(τ+σ) (3.29)

Concerning internal coordinate, following the same steps as in bosonic string case we can find mixed boundary conditions. The generalization of (3.21) reads:

Zk(τ, σ) = zk+ i α
2 n∈Z
zn_+k n + k e−i(n+k)(τ−σ)−z−n−k n + k ei(n+k)(τ+σ)  (3.30)

The complex fermionic coordinates Ψk= ψ2k+iψ₂ 2k+1 become:

Ψk₋= √1 2 r∈Z+a Ψµ_r+ke−i(r+k)(τ−σ) Ψµ₊= √1 2 r∈Z+a ˜ Ψµ_r+ke−i(r+k)(τ+σ) _(3.31)

Finally the mass formula:

α
M2 = i=2,3 m>0 : xi_−mxi_m: + i=2,3 r∈Z+a |r| : ψi −rψri : + k=1,2,3 m∈Z : ˜z_mk_+ kz i m+_k : + k=1,2,3 r∈Z+a |r + k| : ˜Ψi −r+kΨ i r+k : +E (3.32)

where, as we will see in next chapter:

E = ⎧ ⎪ ⎨ ⎪ ⎩ 1 2 3 i=1 |k| −1 2 N S 0 R (3.33)

We are interested in scalar particles so we focus on NS sector. The ground state|p, Λ > is a scalar tachyon annihilated by all oscillators with positive modes. However it is projected out by GSO. We can raise vacuum acting with ˜Ψk_−r+ and Ψ_−r+. In the hypothesis that all angle are positive we obtain that lightest states are following three scalars:

˜ Ψ1₋1 2+|p, Λ > 2α
_M2_=− 1+ ₂+ ₃ ˜ Ψ2₋1 2+|p, Λ > 2α
_M2_{= +} 1− 2+ ₃ ˜ Ψ3₋1 2+|p, Λ > 2α
_M2_{= +} 1+ ₂− ₃ (3.34)

All these states are degenerate and massless for parallel stacks of branes and belong to a N=4 vector multiplet together with other three scalars11. Intersecting branes break down N=4 supersymmetry to lower supersymmetries: for general angles no supercharge remains unbroken; however for very special values some supersymmetries can survive.

In our model we take following angles:

₃= 0 ₁ = ₂= φ (3.36)

With our choice first two scalars of (3.34) become massless while third one acquires a mass α
M2 = φ. As will be proved in next pages these particular values for j preserve N=2 supersymmetries. We now want to analyze low energy spectrum and show that it matches with a D=4, N=2 supersymmetric theory. Combining bosonic field from NS sector with fermionic one from R sector we can construct several multiplets:

• masseless hypermultiplet composed by two N = 1 chiral multiplets Φ(φ1, φ₂): φ₁ : Ψ1₋1

2+φ|0 >N S u1|0 >R (3.37)

φ₂ : Ψ2₋1

2+φ|0 >N S u2|0 >R

where the ui|0 >R represent the two four-dimensional spinors that come from dimensional reduction 6→ 4 of the Ramond vacuum.

• massive vector multiplet composed by one N = 1 chiral multiplet and one N = 1 vector multiplet (A, χ):

A : Ψµ₋1 2|0 >N S Ψ 1 −φΨ2−φu1|0 >R (3.38) χ : Ψ3₋1 2|0 >N S Ψ 1 −φΨ2−φu2|0 >R • massive hypermultiplet

If rotation angles of branes are small enought we can consider vector multiplet in low energy spectrum; otherwise we should keep only the massless hypermulti-plet.

Note that these states transform in the bifundamental of U (2)× U(1) since the string has its ends on different stacks.

11_{For parallel branes all states ψ}M_{|p, Λ > are massleless (j=4,. . . ,9). When we rotate a stacks}

we get three lighter state (3.34) and three heavier states:

α
_M2_(Ψ1

−1

3.2.3 Branes and supersymmetries

As already said, our choice of angles preserve N=2 supersymmetries in four di-mensions. This can be justified not only by mass levels matching but also in a more formal way. Let’s start from type IIB theory with a D9-brane and O(9) ori-entifold: this can be regarded as a type I theory since the presence of a stack of brane breaks half supersymmetries. Type IIB supersymmetries are associated to two currents, one right moving and one left moving. Corresponding charges are two Majorana-Weyl spinors of the same chirality. The unbroken supercharges in presence of branes are:

Qα = Qα+ ˜Qα (3.39)

Now we dualize X5, X7, X9 in order to obtain a stack of D6-branes. A T-duality operation corresponds to a reflection in right sector12 than supercharges spinor changes as:

Qα−→ Qα+ (r5r7r9Q)α˜ (3.40)

dove rj implements reflection on spinors. For arbitrary branes the supercharged preserved are:

Qα= Qα+ (r⊥Q)α˜ (3.41)

with r⊥ a reflection in the directions normal to the stack. If now we have two intersecting stacks of D6-branes the preserved supercharges are those that belong both to (3.40) and to:

Qα = Qα+ (e−iπ(1J45+2J67+3J89)r5r7r9eiπ(1J45+2J67+3J89)Q)α˜ (3.42) The solution corresponds to spinors left invariant under (e−iπ(1J₄5+₂J₆7+₃J₈9)₎2_. Their existence is allowed only if latter operator admits 1 as eigenvalue, namely if one of following condition is satisfied:

₁+ ₂+ ₃ = 0 −1+ ₂+ ₃ = 0 ₁− ₂+ ₃ = 0

₁+ ₂− ₃ = 0 (3.43)

When angle are non vanishing only one of previous conditions can be verified, thus only N=1 supersymmetry can be preserved in four dimensions by stacks intersecting in each two-plane. Our choice instead preserves N=2 supersymmetries.

12_{Remind X}i

3.3

Scattering Amplitudes

Computations of scattering events in string theory involves several techniques of conformal field theory and differential geometry. Expectation values of suitable operators called Vertex operators (VO’s) are evaluated through formalism of Path Integral. Before entering in details of how construct a scattering amplitude we want to compare field theory loop expansion with its analogous in string theory: topological world sheet expansion.

3.3.1 Topology of world-sheets

We come back for a moment to bosonic string action. Even though usually we don’t mansion it, there is one supplementary term which can be added to equation (3.1) and that preserves invariance under two-dimensional diffeomorfism, d-dimensional Poincar´e and conformal transformations:

S
= SB+ λ 

Md

2_σ√_{−γ R}(2)_{(γ) (3.44)}

where γab is the two dimensional metric on the manifold M and R(2) the Ricci curvature. This term however represents a topological invariant for the world-sheet and is proportional to Euler number χ:

χ = 2− 2g − b − c (3.45)

where g is the genus of the manifold, b the number of boundaries and c the number of crosscaps13.

Figure 3.2: Topological expansion for a closed string four points amplitude. Next to leading order are given by closed surfaces of increasing genus.

Path integral on string theory contains a sum over all possible world-sheet manifolds. Adding this piece to the action translates in weighting all correlation function by a factor:

VO’s ∼

M

e−λχ(M) VO’s_S

B(M) (3.46)

13_{For what concerns our discussion c will always be 0. Simple manifold with crosscaps are}

If λ is negative this expansion becomes a perturbative serie whose leading terms are those with higher Euler number.

It’s clear (see for example figure(3.2)) that closed string dynamic is described by closed world-sheets while open string ends create boundaries in the manifold.

In our discussion we deal with leading terms of open strings scattering so we need to analyze first terms of expansion (3.46). The simplest topology in order to describe open strings scattering events is given by manifolds with disk topology. D2 can be interpreted like a sphere cup, that is a g = 0 manifold with b = 1 and therefore χ = 1. We recall a results of Riemann that shows the equivalence of all such manifolds since using conformal transformations plus diffeomorfisms a generic metric can be brought in standard form. As consequence the sum over this topology reduces to one term but with a degeneration over all possible metrics.

a _a a 2 a3 h−1 b₃ bh−1 b₁ 1 2 b α₁ α₂ α₃ α_h

Figure 3.3: Two dimensional surfaces with b boundaries describing open string amplitude can be regarded as closed surfaces with genus g = b− 1.

Completely different is the case of next contribution: annulus or cylinder. We can regard it as a sphere without two caps (g = 0 and b = 2 than χ = 0), however for higher genus surfaces it’s worthwhile consider the closed manifold obtained attaching a mirror copy to the original one (see figure (3.3)). In the case of an annulus we get a Torus and we can easily come back to cylinder with a suitable projection. Contrarily to the sphere, the torus admits an infinity of inequivalent metrics, but they can be labelled just by a complex parameter τ . With conformal mapping a torus can be sent into a complex plane in two ways:

T2 → C/(z ∼ z + 2nπ + i2mπ) ds2 =|dσ₁+ τ dσ₂|2

T2 → C/(z ∼ z + 2nπ + τ2mπ) ds2=|dσ₁+ idσ₂|2 (3.47) We will use second mapping for the whole discussion. The modulus τ defines the equivalence lattice in the complex plane.

Topological expansion in string theory corresponds to a perturbative expan-sion as loop expanexpan-sion in field theory. Looking at figure (3.2) we can regard the scattering as mediated by loops closed strings. The analogy is not at formal level but can be made quantitative when we consider the α
−→ 0 limit (low energy

limit). In this context we recover a real QFT: N-points amplitudes calculated using string formalism at 0 order in α
and at genus g sum up all Feynman diagrams that contribute to that scattering at g loops.

3.3.2 Insertion of VO’s

As discussed in previous section, a scattering amplitude can be regarded as a cor-relation function evaluated on a two dimensional manifold over which are attached asymptotic incoming and outcoming string states. Since bosonic string action con-tains only holomorphic and anti-holomorphic fields ∂X, ¯∂X correlation functions are completely characterized by the behavior of latter operators near punctures (points of asymptotic state insertions). Thus in and out states can be substituted by suitable operators that reproduce the same boundary conditions. These are called Vertex Operators.

Let’s consider a state |s > on the world-sheet and apply to it the operator:

∂Xµ=−i α
2 m αµ m zm+1 (3.48)

Using Cauchy theorem we can obtain the action a single oscillator on the state |s >: αµn|s >= i  2 α
 dz 2πiz n_∂Xµ_{(z)|s > (3.49)}

Suppose that the VO s(z, ¯z) is associated to the state |s >: the Taylor-Laurent expansion of the product ∂Xµ(z) s(w, ¯w) for z ∼ w is called Operator Product Expansion (OPE) and reads:

∂Xµs(w, ¯w) =−i α
2 m αµ_m zm+1 s(z− w, ¯z − ¯w) (3.50) Thus we can infer the association:

αµ_ms(w, ¯w) ←→ αµm|s > (3.51)

and inverting (3.50) we obtain the explicit expression.

We can consider the action of energy momentum tensor T(z) on such VO: T (z) =−1 α
: ∂X ¯∂X : (z) =− 1 α
m Lm zm+1 (3.52)

and again inverting through Cauchy theorem the action of Lm on s(z, ¯z). We remind that condition for a state to be physical reads:

and so is for the associated VO. Operators satisfying such condition are called Conformally Prime Operators and under z−→ z
(z) transform as:

s(z, ¯z)−→ s
(z
, ¯z
) =
∂z
∂z _−h
∂ ¯z
∂ ¯z _−¯h s(z, ¯z) (3.54) where the Conformal weights h, ¯h are defined by the OPE:

T (z) s(0, 0) = h

z2s(0, 0) +

∂s(0, 0)

z (3.55)

and analogous with ¯T (¯z), ¯h, ¯∂. Position of VO’s must be integrated over the whole word-sheet14thus in order to keep conformal invariance unbroken we must choose VO conformal weights (h, ¯h) = (1, 1) so that:

 d2z s(z, ¯z)−→  d2z
∂z
∂z −h+1
_{∂ ¯z}
∂ ¯z −¯h+1 s(z, ¯z) (3.56) and correlation functions are conformal invariant.

Let’s conclude this section considering VO for the bosonic closed string ground state:

|0, k >=: eik·X _{: (3.57)}

It’s annihilated by all oscillator with positive mode number while Conformal weights furnish the mass shell condition: 1 = k2₄α
.

3.3.3 Conformal Killing vectors and Pictures

As already discussed strings amplitudes are evaluated using path integral for-malism. General techniques familiar in Field Theory can be adapted to present context, such as the introduction of ghost sector in order to eliminate degrees of freedom. Let’s start from the bosonic case: path integral involves integration over all possible metrics of the manifold; bosonic action, however, admits a large group of symmetries thus effective degrees of freedom are given by coset space:

S = G

(Diff)× (Weyl) (3.58)

Diffeomorphisms plus Weyl rescaling do not manage to bring a metric in unique form but may exists several classes of inequivalent metrics labelled by parameters called moduli15. The space S is also called moduli space.

In previous section we have described how a scattering amplitude is charac-terized by insertions of Vertex Operators to simulate incoming and outcoming states. Variables σi, on which these operators are inserted, are integrated over whole manifold. As consequence parameters space must be enlarged:

14_{The boundary of world-sheet in the case of open string} 15_{For torus the modulus is complex parameter τ}

S
₌ G × Mn

(Diff)× (Weyl) (3.59)

At this point it becomes possible using all symmetries of the action to reduces integration variables. Indeed some manifolds admit particular combinations of Diffeo plus Weyl transformations that leave unchanged the metric, called Confor-mal Killing Vectors. In space (3.58) these symmetries get lost while can be used in spaceS
to fix some coordinates to precise values. Taking all in account we can introduce a ghost sector and transform integrals ([14]):

 [dg] n  i=1  Md 2_σ i (3.60)

in a misure over moduli space plus a ghost determinant ∆g:  Sdτ n  j=(k+1)  Md 2_σ j  [dbi][dci] k  i=1 c(xi)∆g (3.61)

where τ represents all moduli, b, c are ghost fields and xi are arbitrary points on the manifold M. The number of CKV’s k is given by an important result of Riemann and Roche connecting it to Euler number χ and moduli number µ

χ > 0 ⇒ k = 3χ, µ = 0

χ = 0 ⇒ k = µ

χ < 0 ⇒ k = 0, µ = −3χ (3.62)

In presence of supersymmetry the discussion is more subtle. We limit here to present the prescription and we refer to [13] for a detailed discussion. We define the Picture of a Vertex Operator representing bosonic or fermionic states respectively an integer or half-integer charge q:

s(z, ¯z)−→ e−qφs(z, ¯z) (3.63)

where φ is a free scalar field. Latter field comes from the bosonization of a ghost field. Normally the value of q is−1 or −1₂, however can be increased of one unity acting on the VO with the Picture Changing Operator (PCO) defined in equation (4.20). Total Picture charge of an amplitude is fixed by an extension of Riemann-Roche theorem to be 2g−2 thus one has to insert an appropriate number of PCO’s in order to compensate charges in excess16.

3.4

Conformal Field Theory

3.4.1 Disk

The first calculation that we present will concern tree level scattering of four open strings. According to topological expansion leading contribution is given by a disk-like world-sheet. Such a manifold can be mapped in upper half-plane with a conformal transformation while boundary is sent into real axis. The Euler number of the surface is equal to one, thus we have:

• 3 CKV’s

• No modulus: all metrics are equivalent • Total Picture charge = −2

Vertex Operators are inserted on real axis: three of them are fixed in given positions while forth one must be integrated Moreover we have to transform two VO’s from −1-picture to 0-picture.

3.4.2 Torus and Annulus

Next to leading order is represented by Annulus, however we discuss the torus case first. Topology gives:

• 2 CKV’s

• 2 real moduli: the complex parameter τ • Total Picture charge = 0

Notice that, since this surface emerges in closed strings scattering, VO’s are in-serted in the ”bulk”; thus in order to fix their positions two CKV’s are necessary for each. In order to perform calculations on this surface one needs expressions of elementary fields correlation functions. For bosonic fields all formula are given in appendix B.

Thinks are more complicated for fermionic fields. As already discussed Super-string Theory is a two dimensional conformal theory that makes sense only in ten dimension. Only for this particular value the theory is Poincar´e-anomaly free. The demonstration, however, is based on the assumption of modular invariance. The request that latter holds is than a consistency condition for string theory.

After this remark let’s come back to torus amplitude. Genus equal to one implies the existence of two inequivalent homotopy classes; we call α, β two rap-presentative curves of these classes. Closed strings can wrap along these curves, thus one needs to specify carefully boundary condition. Let’s focus on right moving fields: there are four cases, generally indicated by:

[a, b] a, b =±1

2 (3.64)

depending on the periodicity of fermionic coordinate along curves α, β. Note that only one of this periodicity is related to R,NS boundary conditions (let’s say a): one can choose a cycle to represent periodicity along σ−→ σ + 2π. Other parameter (b) indicates periodicity in τ . Closed strings admit also left moving sector for a total of sixteen cases.

It can be shown that modular invariance of Patition function requires a sum over these four Spin Structures with suitable coefficients. The same sum must be performed for generic amplitudes on the torus as well.

We remind that a closed string admits two sets of indipendent oscillators: we focus on right-moving fields only. This is equivalent to what we obtain considering an open string.

Correlation function of fermionic coordinates are better evaluated passing in bosonized description; let’s define:

: ¯ψψ :=: eiH : (3.65)

It can be proved that latter formulation is completely equivalent to original one considering the OPE with all other fields of the theory. Not all fermionic fields admit such simple bosinization: in general for twisted fields we have:

: ¯ψψ :=: eiγH : γ ∈ R (3.66)

We are now able to write down the general formula for fermionic correlation function in a spin structure [a, b]:



: eiγH(x) :: e−iγH(y) :  =
θ₁
(0|τ) θ₁(γ(x− y)|τ) γ2 1 η(τ )θ  a b  (γ(x− y)|τ) (3.67) In the case of an annulus we can stop here; for a torus we would have an equal expression for left-moving fields.

We can finally introduce the annulus: • 1 real CKV

• 1 modulus

• Total picture charge=0

The cylinder can be obtained starting from a torus with an appropriate involution. We start consider mapping (3.47) with imaginary modulus (τ = it) and we identify points through the equivalence: z ∼ −¯z + π. Latter identification introduces two boundaries in the fundamental region of the lattice.

This time only one CKV is present: however it is enough to fix one position of VO’s, since for open strings they must be inserted on boundaries of a two-dimensional surface.

Our computation will involve VO’s of four bosons: since they naturally emerge in−1-picture they all must be transformed to 0-picture.

Four points amplitude

In the same way as in Quantum Field Theory, the 0-order term of string loop expansion of the H†HF2 coefficient is vanishing. This is proved is two different ways: firstly in section 4.1 Higgs mass corrections due to the presence of an external magnetic field are computed and the absence of a quadratic dependence from the latter is shown. Secondly, after having discussed the form of Higgs and gauge vector Vertex Operators (VO’s) in section, a brief description of Higgs self-energy on disk is given in section 4.3.

Than we move on first order in string loop corrections. The general frame of one loop amplitudes is described in 4.4, while the explicit calculation of the four points amplitude hh → AA is carried out in section 4.5. The presence of a non vanishing contribution is found in the N = 1 supersymmetric case.

4.1

External Magnetic Field

We solve the equation of motion of an open string in presence of constant magnetic field. Than we compute Virasoro algebra and we extract the central charge of this particular CFT, in order to obtain the vacumm energy dependence from external field. We find a linear dependence plus cubic order corrections.

4.1.1 Open string in background magnetic field

We start considering an open string interacting with an external magnetic field; the world sheet action reads:

S = − 1 4πα
 dσdτ ∂aXµ∂aXµ− i ¯ψµρa∂aψµ  +qL 2  dτ Fµν
Xν∂τXµ− 1 2ψ¯ ν_ρ0_ψµ  σ=0 +qR 2  dτ Fµν
Xν∂τXµ−1 2 ¯ ψνρ0ψµ  σ=π (4.1)

Bulk terms bring usual free wave equations for Xµand ψµ, while second and third one modify boundary conditions for previous fields. Let’s focus on the bosonic coordinates:

∂σXµ = πqRFνµ∂τXν (σ = 0)

∂σXµ = −πqLF_νµ∂τXν (σ = π) (4.2)

We restrict to a pure constant magnetic field in one of spacial directions (F₂₃= M ) and we choose the gauge:

A₃= M X2 (4.3)

It is worthwhile combining these two coordinates in a complex fields: Z₁ = √1

2(X

2_{+ iX}3_{) Z}¯_{1 = √}1 2(X

2_{− iX}3_{) (4.4)}

Boundary conditions take the form:

∂σZ1 = −iβR∂τZ1 (σ = 0)

∂σZ1 = iβL∂τZ1 (σ = π) (4.5)

where β_R(L) = πM q_R(L). We consider only the case qL+ qR= 0, since the case of neutral Higgs is trivial, as will be shown in section 4.3. Mode expansion for the complex bosonic coordinate is ([8]):

Z₁ = z + i ∞ n=1 znφn− i ∞ m=0 ˜ z_m†φ_−m (4.6) where: zn=|n − M|1/2cos ((n− M)σ + γL)e−i(n−M)τ (4.7) Here we have defined:

γR(L) = tan−1(βR(L)) M = 1

π(γL+ γR)

We now study the Virasoro algebra of the theory. Virasoro operators Lm are defined as usual to be Furier components of the energy-momentum tensor:

Lm =  C dz 2πiz m+1_T zz (4.8)

where C is any contour encircling the point z = 0 counterclockwise. After a straightforward calculation ([7]) we obtain the Virasoro algebra:

[Ln, Lm] = (n− m)Ln+m+
2 12(n 3_{− n) + nM(1− M)}_δ n+m (4.9)

The result is what we expect apart a new term in the central charge linear in n that can be eliminated shifting L₀ → L₀+ 1₂M(1− M). This corresponds to changing the normal ordering constant of a quantity−1₂M(1− M).

Let’s come back to the fermionic sector of the theory. To derive the right boundary conditions we impose ([15]):

δψ_Rν = δψ_Rµ (σ = 0)

δψ_Rν =−(−1)aδψµ_R (σ = π) (4.10)

where ψR(L) are the right(left)-moving components of the fermion and a = 0(1) represents NS(R)-sector. Boundary conditions read:

ψ_Rµ− ψ_Lµ= πqRF_νµ(ψ_Rν + ψ_Lν) (σ = 0)

ψ_Rµ+ (−1a)ψ_Lµ=−πqLF_νµ(ψ_Rν − (−1)aψ_Lν) (σ = 0) (4.11) In the case in which only F₂₃= 0 the mode expansion for the complex combination Ψ₁= √1 2(ψ2+ iψ2) gives ([8]): Ψ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∞ s=1/2 Ψsfs+ ∞ m=1/2 ˜ Ψ†_mf_−m N S ∞ n=1 Ψnfn+ ∞ m=0 ˜ Ψ†_mf_−m R (4.12) with: f(R L),n = √1 2e

−i(n−M)(τ∓σ)±i tan−1(βL) _(4.13) and clearly fn = fR,n+ fL,n. In the same way as in the previous case we can extract the shift δcF in the fermionic normal ordering constant:

δcF =

1

22M N S

+1₂(M − 1)M R (4.14)

4.1.2 Spectrum of open strings at the intersection

We remind results obtained in previous chapter for the spectrum of an open string stretching between two intersecting stack branes in N=2 configuration:

α
M2 = NSpacetime+ NCompact Space+ E (4.15) where:

E =

φ−1₂ N S

0 R (4.16)

E =

φ−1₂ +1₂M N S

0 R (4.17)

Presence of an external magnetic field breaks both supersymmetries: the bosonic and fermionic component of the chiral multiplets are no more degenerate. In par-ticular the two Higgses acquire a mass:

2α
M2 = 1 π  tan−1(πqRM ) + tan−1(πqLM )   (qR+ qL)M + O(M3) (4.18)

in the limit of small external field.

A tree level interaction of the form H†HF2 would bring a quadratic depen-dence from the vev M in the mass expression. However equation (4.18) proves the absence of such interaction.

4.2

Vertex Operators

We construct Higgs VO using formalism of CFT on orbifolds. We introduce bosonic and fermionic twisted fields. Than we construct Higgs complex conjugate VO and we pass in 0-picture applying the Picture Changing Operator (PCO).

4.2.1 Gauge vector Vertex Operator

Gauge vectors VO in−1-picture reads:

V_A−1(z, k, ξ) =√2α
goe−φ(ξ· ψ)eik·X (4.19) where ξµ is four-dimensional polarization vector while kµ is the dimensional mo-mentum. The constant go is related to the λ that labels string loop expansion and are background dependent as will be discussed in next chapter . As already discussed, the PCO is given by a superconformal ghost factor eφ times the total supersymmetry current: P = eφTF(z) = i  2 α
e φ a ψa∂Xa(z) (4.20)

Applying it to V_A−1 we get gauge vertex operator in 0-picture: V_A0(z, k, ξ) =  2 α
go  iξ· ∂X + α
(ξ· ψ)(k · ψ)eik·X (4.21)

4.2.2 Higgs Vertex Operator

Twisted fields

We have seen in previous chapter how two intersecting D6-branes can be inter-preted as D9-branes with constant magnetic background and three coordinates dualized. Let us start with an equation similar to (4.2) for Zk= √1₂(X2k+iX2k+1), k = 2, 3, 4 and let’s dualize along X2k+1direction. Now boundary conditions read:

Re(∂σZk) = 0 Im(Zk) = 0 (σ = 0)

Re(e−ik_{∂σZk) = 0 Im(e}−k_{Zk) = 0 (σ = π)} (4.22) where krepresent angles between branes in the planes (X2k, X2k+1). As in section 4.1 these boundary conditions modify mode expansions shifting mode numbers by

k

π. It follows for holomorfic fields:

∂Z(e2πiz) = e2ik_{∂Z(z) ∂ ¯Z(e}2πi_{z) = e}−2ik_{∂ ¯Z(z) (4.23)} that is to say latter fields have a cut at the point z = 0. This is exactly the same behavior of fields living on orbifolds; conditions (4.23) are implemented by means of operators called twist fields([11]) that have appropriate OPE:

∂Zj(z)σj₊(w)∼ (z − w)−(1−j)_τj

+(w) (4.24)

∂ ¯Zj(z)σj₊(w)∼ (z − w)−j_τ
j

+(w) (4.25)

Superconformal symmetry requires a similar twisting for world-sheet superpart-ners as well; in NS sector we must introduce twist fields S+_k:

Ψj(z)S₊j(w)∼ (z − w)j_t
j

+(w) (4.26)

¯

Ψj(z)S₊j(w)∼ (z − w)−j_tj

+(w) (4.27)

where we have defined Ψk= √1₂(ψ2k+ iψ2k+1). In the bosonic description:

Ψj = eiHj _Ψ¯j _{= e}−iHj _(4.28)

it follows for twist fields:

S₊j = eijHj _t
j

+ = ei(j+1)Hj tj+= ei(j−1)Hj (4.29)

Their conformal weight are:

[σ₊] = [σ₋] = φ

2(1− φ) [τ₊] = [τ₋
] = φ

[τ₊
] = [τ₋] = φ 2(1− φ) + (1 − φ) [eiβH] = 1 2β 2 _(4.30) H and H†

NS ground state VO for intersecting branes in N=2 configuration reads:

V₋₁0 = goe−φΛ(σ₊1S₊1)(σ₊2S₊2)eip·X(z) (4.31) where Λ is the Chan-Paton factor and the twist fields implement the change of boundary in passing from one brane to another.

In case of positive intersections the lightest scalars are obtained applying ¯

Ψk₋1

2+k to NS vacuum. For instance, one of Higgses VO in −1-picture is given by:

V_h−1(z, p) = 

dw(w− z)−(1+j)_Ψ(w)V0

h = goe−φΛ(σ₊1S₊1)(σ₊2t2₊)eip·X(z) (4.32) The C-conjugate Higgs is obtained considering the string with opposite ori-entation. This in practice is equivalent to replace φ ↔ (1 − φ) in expressions (4.24),(4.25) and to change the signs of angles in (4.26),(4.27),(4.34):

∂Zj(z)σj₋(w)∼ (z − w)−j_τj −(w) ∂ ¯Zj(z)σj₋(w)∼ (z − w)−(1−j)_τ
j −(w) Ψj(z)S₋j(w)∼ (z − w)−j_t
j −(w) ¯ Ψj(z)S₋j(w)∼ (z − w)j_tj −(w) (4.33) with: Sj₋= e−ijHj _t
j −= ei(−j+1)Hj tj−= e−i(j+1)Hj (4.34) However we should take care to the fact that for negative intersections the lightest states aren’t ¯Ψk

−1

2+k|0 >N S but are Ψ k −1

2+k|0 >N S. The effect of this is that in −1-picture Higgs VO doesn’t appare tk

− = e−i(k+1) but t
−k = ei(−k+1) instead. Summarizing: V_h−1 1 (z, p) = goe −φ_Λ(σ1 +σ+2)ei1(H1+H2)−iH2eip·X(z) V_¯h−1 1 (z, p) = goe −φ_ΛT_(σ1 −σ−2)e−i1(H1+H2)+iH2eip·X(z) V_h−1₂ (z, p) = goe−φΛ(σ₊1σ₊2)ei1(H1+H2)−iH1eip·X(z) V_¯h−1 2 (z, p) = goe −φ_ΛT_(σ1 −σ−2)e−i1(H1+H2)+iH1eip·X(z) (4.35)