Revisiting Higher-Spin Gyromagnetic Couplings
aa r X i v : . [ h e p - t h ] F e b Revisiting Higher-Spin Gyromagnetic Couplings
Raffaele Marotta a , Massimo Taronna a,b and Mritunjay Verma c a Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli,Complesso Universitario di Monte S. Angelo ed. 6, via Cintia, 80126, Napoli, Italy. b Dipartimento di Fisica “Ettore Pancini”, Universita’ degli Studi di Napoli,Federico II, Monte S. Angelo, Via Cintia, 80126 Napoli, Italy c Mathematical Sciences, Highfield, University of Southampton, SO17 1BJ Southampton, UK
E-mail: raff[email protected], [email protected], [email protected]
Abstract
We analyze Bosonic, Heterotic, and Type II string theories compactified on a generic torushaving constant moduli. By computing the hamiltonian giving the interaction between massivestring excitations and U (1) gauge fields arising from the graviton and Kalb-Ramond field uponcompactification, we derive a general formula for such couplings that turns out to be universal inall these theories. We also confirm our result by explicitly evaluating the relevant string three-point amplitudes. From this expression, we determine the gyromagnetic ratio g of massive stringstates coupled to both gauge-fields. For a generic mixed symmetry state, there is one gyromagneticcoupling associated with each row of the corresponding Young Tableau diagram. For all the stateshaving zero Kaluza Klein or Winding charges, the value of g turns out to be . We also explicitlyconsider totally symmetric and mixed symmetry states (having two rows in the Young diagram)associated with the first Regge-trajectory and obtain their corresponding g value.1 ontents T D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Type II and Heterotic strings on T D . . . . . . . . . . . . . . . . . . . . . . . . . 14 p R = ± p L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Gravitational minimal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Totally symmetric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Mixed-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A.1 Bosonic sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2 Type II sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.3 Heterotic sigma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B Generalities on higher spin states 36C Summary of open string amplitudes for the bosonic states in the first Reggetrajectory 38
C.1 Generating Functions for Vertex operators . . . . . . . . . . . . . . . . . . . . . 38C.2 Summary of open string amplitudes for the bosonic states in the first Regge trajectory 39
D Young Tableaux and Polynomials 44 Introduction
Charting the landscape of gravitational theories is a problem of key importance both to achieve abetter understanding of string theory as well as to clarify the basic consistency requirements thatgravitational theories should abide to. While the attention is often restricted to massless fields ofspin up to two, a generic feature of consistent gravitational theories is the presence of infinite towersof massive fields of increasing spin.In this work we focus on the leading electromagnetic coupling of massive excitations which isthe first non-minimal correction to the minimal coupling. The problem of understanding how tocouple consistently such massive excitations has a long history starting from Fierz and Pauli [1] whowere the first to analyse consistent electromagnetic interactions of massive spinning fields. Theproblem turned out to be more subtle than expected since just introducing a minimal coupling wasnot a consistent solution. It was quickly realised that minimal electromagnetic interactions must besupplemented by a non-minimal coupling proportional to the Electromagnetic tensor F µν and whoseoverall coupling constant is usually expressed in terms of the electric charge q up to a numericalfactor usually referred to as gyromagnetic ratio g . The gyromagnetic ratio determines the intensityof their magnetic moments in the interaction with external magnetic fields.In the ‘50s, using the minimal coupling, Belinfante hypothesized that g had to be equal to theinverse of the spin of the particle [2]. This relation gives the correct value for a Dirac particle butit fails to give the correct tree level gyromagnetic ratio of the the other higher spin (HS) particlessuch as W-boson which turns out to be g = 2 . This is expected since, as mentioned above, theminimal coupling does not give rise to a consistent electromagnetic interaction of the higher spinfields.The value g = 2 naturally arises also from a number of different approaches which includethe study of the precession of a spin particle in a magnetic field [3], the high energy behaviourof scattering amplitudes [4] and string theory [5, 6, 7, 8] that suggest that the natural value forthis ratio is g = 2 (see also Ref. [9] for a review on the subject). Following these results, in the90s, Ferrara, Porrati and Telegdi proposed an electromagnetic coupling which is consistent with thevalue g = 2 for all the elementary particles of arbitrary spin [7]. The value g = 2 has also beenrecently clarified in the context of massive HS interactions [10, 11, 12]. It turns out that consistentpropagation on constant electromagnetic backgrounds (in the absence of other fields) are possibleonly with g = 2 . However, the analysis in [11] left open the possibility of other values for g in morecomplicated theories with additional fields. Similar analysis has also been performed by imposingcausality and no superluminal propagation in constant electromagnetic background leading to thesame value g = 2 [13, 14, 15]. Again, however more complicated cases with additional fields havenot been yet analysed from this perspective.In fact, there are some exceptions to this natural value of the gyromagnetic factor. In [16], itwas shown that the massive spin-two particles, arising from compactification of the five-dimensional3instein-Hilbert action, have g = 1 . The same value was obtained via soft-theorem compactifica-tions of d + 1 dimensional theory to d dimension in [17]. The D0 branes in Type IIA theory, whichare the KK modes associated to the circle compactification of D = 11 SUGRA, were also explicitlyshown to have g = 1 in [18].In this draft, motivated by these results, we closely analyse the gyromagnetic factor for massivestring excitations with respect to the U (1) gauge fields arising from both the Graviton and KalbRamond fields upon compactification. We clarify how the value g = 2 for open string states in 10dimension is a simple consequence of conservation of the corresponding vertex operator. For modeswith vanishing winding charge, the g = 1 value for closed string states, turns out to arise as a simpleconsequence of the structure of the Graviton vertex operator that is written as a double-copy ofopen string vertex operators. Thus, for all string excitations with a field theoretic interpretation,the value g = 1 simply follows from the consistency of minimal coupling before the reduction.More specifically, we shall consider the compactification of Bosonic, type II and Heterotic stringtheories on general toroidal backgrounds in the presence of the Graviton and Kalb Ramond fields.We shall keep the internal components of these fields (which are related to moduli) to non-zero butarbitrary constant values. Following the method used in [19] to determine the Hamiltonian of thebosonic string compactified on circles, we shall read the gyromagnetic ratios from the expressionsof interacting Hamiltonians which describe the interaction between gauge fields and the massive HSfields under compactification. The same results will also be obtained from an amplitude calculationperformed by considering a momentum expansion of the massless string vertex. The final resultabout the electromagnetic coupling of string excitations to the first order in the field expansion andderivatives, can be written in terms of the minimal coupling plus non-minimal terms proportionalto the internal spin operators S µνR ; L , and it is invariant in form for all the string theories aboveconsidered. It turns out to be: V ( A,B )ΦΦ ≃ D Φ (cid:12)(cid:12)(cid:12) ( p aL + p aR ) A a · p − F Aµν ; a ( p aL S µνR + p aR S µνL )+ ( p aL − p aR ) B a · p − F Bµν ; a ( p aL S µνR − p aR S µνL ) (cid:12)(cid:12)(cid:12) Φ E (1.1)where F A ; Bµν ; a are the field strengths of the gauge fields arising from the metric and Kalb-Ramondcompactification respectively, while p aR ; L are the compact momenta and p is the momentum of thehigher-spin states.The states with field theoretic interpretation turn out to have g = 1 as mentioned above.However, the compactification also gives rise to states characterized by Kaluza Klein and Windingcharges. These are the charges with respect to the gauge fields arising due to the compactificationof the Graviton and Kalb Ramond fields respectively. It turns out that the vanishing of either ofthese charges corresponds to states with g = 1 . On the other hand, the general twisted sectorstates have gyromagnetic ratios which depends upon the KK and Winding charges as well as spins.While these cannot be predicted by analysing minimal gravitational coupling and its dimensional4eduction, it is clear that their pattern is also fixed by the double-copy structure of the gravitonvertex operator.One subtlety associated with the higher spin fields is regarding their symmetry properties. Thegeneral string states have mixed symmetry polarisation tensors which are characterised by a genericYoung Tableau diagram. However, in the literature, these states have not received much attention.Our analysis shall also be applicable to these states. In particular, we give explicit results for themixed symmetry states described by Young tableau diagrams having two rows from first Reggetrajectory of string spectrum. It turns out that there is one gyromagnetic coupling (and hencea gyromagnetic ratio) associated with each row of the Young Tableau diagram describing thecorresponding string state. One of our main result is the following gyromagnetic factors g ( a )1 = 2 p aL − p aR ( ℓ R − k ) p aL − ( ℓ L − k ) p aR ℓ R + ℓ L − k , (1.2) g ( a )2 = 2 p aL − p aR − ( ℓ R − k ) p aR + ( ℓ L − k ) p aL ℓ R + ℓ L − k , (1.3)for the mixed-symmetry states in the first Regge trajectory of closed string theories with respectto the gauge boson obtained from compactification of the metric. In the above expression, the ℓ L and ℓ R denote the left and right spins respectively. The state is represented by an Young diagramwith ℓ L + ℓ R − k boxes in the first row and k boxes in the second row (see figure 2). The firstgyromagnetic factor coupling is associated to the first row while the second to the second row.The rest of the draft is organised as follows. In section 2, we discuss some generalities aboutthe gyromagnetic coupling and introduce some notations which will be useful in our computations.In section 3, we compute the part of interacting Hamiltonian of the compactified Bosonic, type IIand Heterotic theories which encode the information about the gyromagnetic ratios. In section 4,we shall directly compute the 3 point vertices of gauge fields and the massive string states underthe compactification using the string amplitudes. In section 5, we shall read the g factor for somemassive states using the results of section 3 and 4. Finally, we shall conclude with some discussionin section 6. Details of some computations will be given in appendices. In this section, we shall review the concept of gyromagnetic ratio. It will be convenient for thepurposes of the present draft to define the gyromagnetic ratio as a coupling constant appearingin the effective action. In particular, considering the electromagnetic coupling of a field Φ , at thelowest derivative order we can write down two type of couplings: minimal coupling which definesthe charge of the field Φ and it is therefore uniquely characterised by this property and non-minimal For an interesting instance of two gyromagnetic ratios in the context of black holes in Heterotic string theoryon torus, see [20]. F µν V A ΦΦ ∼ iqA µ [Φ ∗ · ( ∂ µ Φ) − ( ∂ µ Φ ∗ ) · Φ] + iα F µν (Φ µ · Φ ν ) . (2.1)Above we have been schematic with the “ · ” implying certain index contractions among the fieldwhich in principle can have an arbitrary tableaux shape. We have also canonically normalised thekinetic term as L = Φ ⋆ · ( (cid:3) + m )Φ , (2.2)assuming the index contraction is of weight in the permutation of the various indices. With theseconventions one can define the gyromagnetic factor as g ∼ (cid:12)(cid:12)(cid:12)(cid:12) αq (cid:12)(cid:12)(cid:12)(cid:12) . (2.3)Note that the gyromagnetic ratio is ill-defined if the charge q vanishes. However one can still definethe coupling α . The above definitions can then be related to the magnetic moment of a particle in4d. The main simplification is that in 4d all particles can be classified as totally symmetric fieldswhich implies the existence of a unique non-minimal electromagnetic coupling of the type (2.1) V N.M.A ΦΦ = iα F µν Φ ∗ µµ ( s − Φ ν µ ( s − , (2.4)Using a generating function notation Φ( u ) = s ! Φ µ ( s ) u µ ( s ) , where we have introduced the totallysymmetric product of u ’s as u µ ( s ) = u µ · · · u µ s , the above equation can also be equivalently writtenas V N.M.A ΦΦ = iα F µν h Φ | S µν | Φ i , (2.5)where we have conveniently introduced the spin operator S µνu = u µ ∂ νu − u ν ∂ µu together with theinner product defined as: h Φ | Φ i = exp ( ∂ u · ∂ u ) Φ ( u )Φ ( u ) (cid:12)(cid:12)(cid:12) u i =0 . (2.6)The manifest appearence of the spin operator clarifies the relation between the angular momentumand the associated magnetic moment. E.g., for a classical massive electrically charged particle withmass m and charge q , the magnitude of the magnetic moment µ and the angular momentum L are related as (see, e.g. [21]) µ = α m L . (2.7)Considering the generic dimensional case the story for totally symmetric representation is unchanged.However, in d > there exist more general representations of the Lorentz group and the number ofnon-minimal electromagnetic couplings can increase. In particular we have as many gyromagnetic6actors as the number of rows of the fields. This also correspond to the number of spin operatorswhich is equal to the number of rows. For concreteness, representing a generic mixed-symmetryfield as a generating function in terms of auxiliary vector variables u i Φ( u i ) = 1 s ! · · · s n ! φ µ ( s ) µ ( s ) ...µ n ( s n ) u µ ( s )1 · · · u µ ( s n ) n , (2.8)subject to the irreducibility conditions u i · ∂ u i + k Φ = 0 for all i and k > , we can define thespin-operators: S µνu i = u µi ∂ νu i − u νi ∂ µu i . (2.9)One is then naturally led to the following basis of gyromagnetic factors: V ( j ) A ΦΦ = i α ( j ) F µν (cid:10) Φ | S µνj | Φ (cid:11) . (2.10)A natural generalisation of the totally-symmetric gyromagnetic factor to mixed symmetry fields isobviously the one that involves the total spin operator V A ΦΦ = i α F µν D Φ (cid:12)(cid:12)(cid:12) n X j =1 S µνj | {z } S µν (cid:12)(cid:12)(cid:12) Φ E . (2.11)In the following we shall focus on extracting the above coefficients α ( j ) in string compactifications. In this work, we shall often need to translate results from String Theory to extract the gyromagneticfactor. In order to do so, it is convenient to introduce a dictionary between string calculations interms of oscillators and the auxiliary variables u i introduced above.While in bosonic string theory one works with states of the form | φ i = N φ µ ( s ) ...µ p ( s p )¯ µ (¯ s ) ... ¯ µ q (¯ s q ) α µ ( s ) − n . . . α µ p ( s p ) − n p ¯ α ¯ µ (¯ s ) − ¯ n . . . ¯ α ¯ µ q (¯ s q ) − ¯ n q | , ¯0 , p i (2.12)expressed in term of α -oscillators, it is often more convenient to work with auxiliary commutingvariables w i sometime referred to as symbols of the oscillators. Performing this step is actuallyextremely simple and can be obtained by replacing the α -oscillators α − n with commuting variables w n as | φ i → s ! · · · s p !¯ s ! · · · ¯ s p ! φ µ ( s ) ...µ p ( s p )¯ µ (¯ s ) ... ¯ µ q (¯ s q ) w µ ( s ) n . . . w µ p ( s p ) n p ¯ w ¯ µ (¯ s )¯ n . . . ¯ w ¯ µ q (¯ s q )¯ n q (2.13)The operator product governing the α oscillators can then be implemented as a certain differentialoperators on the symbols. For instance, the inner product then takes the form (2.6) where u and ¯ u The label u i in S u i will be replaced by a simple index i when no ambiguity can arise. w n are dummy variables and can be changed as needed.In the following it will be useful to also remove the polarisation tensor replacing it with aproduct of vector polarisations. Introducing the vector polarisations u n and ¯ u n one then arrives tothe following representation for the states in (2.12) s ! · · · s p !¯ s ! · · · ¯ s p ! ( u n · w n ) s . . . ( u n p · w n p ) s p (¯ u ¯ n · ¯ w ¯ n ) ¯ s . . . (¯ u ¯ n q · ¯ w ¯ n q ) ¯ s q . (2.14)Focusing on the first Regge trajectory we have only α − oscillators and the general closed stringoperator reads: φ ℓ R + ℓ L = N ℓ R , ℓ L ( u · w ) ℓ R ( u · ¯ w ) ℓ L . (2.15)Here, however, we have a reducible representation. It turns out that using the same auxiliaryvariables, we can also implement conveniently Young projections by working with appropriate poly-nomials which implement the Young projection conditions. For instance, mixed symmetry statesassociated to the Young-Tableaux shown in Figure 2, are constructed in appendix D, by startingfrom the states of the form in Eq. (2.14). Here, we only write the result φ ℓ L + ℓ R − k,k = N ℓ L + ℓ R − k,k ( u · w ) ℓ − k ( u · ¯ w ) ℓ − k ( u · w ¯ u · ¯ w − u · ¯ w ¯ u · w ) k , (2.16)with k an integer number k ≤ Min { ℓ , ℓ } . The expression of such states in terms of oscillators iseasily obtained from the mapping defined in equation (2.13).In this work it will prove very convenient to work with the above polynomials in contractions ofthe auxiliary variables in order to extract the gyromagnetic ratio from the closed string amplitude.We would also like to stress that similar mappings can also be defined for Heterotic or Type IItheories. String theories are naturally defined in the critical space-time dimensions where the conformalanomaly is vanishing. Their spectrum contains massless states identified with the gauge bosons ofthe fundamental interactions and an infinite tower of massive higher spin fields with mass propor-tional to the inverse of the string slope α ′ . They provide a quantum description of the interactionsof such massive high-spin particles with massless and massive states and, therefore, are consistenthigher spin theories with an infinite number of fields.Extracting interesting information about the string theory requires the compactification of severalspatial dimensions. In the compactification, the initial Lorenz group SO ( d − , is broken andnew massless gauge-fields appear in the spectrum. These are the d -dimensional massless tensorfields having one index transforming as a vector under the unbroken SO ( d − D − , and the8ther ones pointing along the compact directions. In the following, we shall focus on the U (1) gauge fields resulting from the compactification of the metric and Kald-Ramond field and we shallcompute the cubic interactions of such gauge fields with the higher spin fields compactified on ageneric torus T D . From these cubic couplings, we can read the gyromagnetic ratios of the high-spinstates charged with respect to these U (1) -fields.In this section, we shall follow the Hamiltonian approach which gives a convenient way toextract the gyromagnetic coupling of massive string states [19]. In this method, we shall computethe expectation value of the Hamiltonian describing the interaction between the gauge fields and theworld-sheet fields. This Hamiltonian, in the case of Bosonic string theory compactified on circles,was considered in [19]. We extend such a result to the Bosonic, Type II and Heterotic string theoriescompactified on a generic D -dimensional compact torus parametrized by the constant backgroundcomponents of the metric and the antisymmetric tensor. Before moving further, we note that forcomparing with the standard expression of Hamiltonian containing the gyromagnetic coupling, itis convenient to multiply the expression of the Hamiltonian with ℓ/ (2 πα ′ m ) and work with thefollowing rescaled Hamiltonian [19] H I ≡ ℓ πα ′ m H I . (3.1) The manifold on which we shall analyse the string theories are generic toroidal manifolds T D . Weshall denote with µ, ν = 0 , . . . d − D − the non compact directions, and with i, j = 1 , . . . D the compact ones. The d will be 26 for the Bosonic theory and 10 for the superstring theories.Since torus is a flat manifold, we can parametrize it so that the metric g ij on it is constant and thecompact coordinates X i have the period π √ α ′ , i.e. X i ( τ , σ + ℓ ) = X i ( τ , σ ) + 2 π √ α ′ n i , n i ∈ Z (3.2)Here, ℓ can be either π or π depending on the adopted conventions. In the above equation, themore general boundary conditions are allowed for the compact directions because the points X i and X i + 2 π √ α ′ n i describe the same point in space-time. In contrast, for the non compact directions,the boundary conditions are X µ ( τ , σ + ℓ ) = X µ ( τ , σ ) (3.3)Instead of g ij , it is convenient to work with the standard Euclidean metric δ ab . This can be doneby introducing a constant vielbein e ai on the torus via g ij = e ai e bj δ ab (3.4) One way to see this is to note that the zero mode of the Hamiltonian in equation (3.18) contains the term H = α ′ πℓ ˆ p + . . . with ˆ p µ being the momentum operator along the non compact directions. Evaluating itsexpectation value for an external state having momenta ( p , ~p ) with mass m and multiplying by ℓ/ (2 πα ′ m ) givesthe standard expression ~p / m of the energy of the free charged particle of mass m . This is also in the agreementwith the expressions in the non relativistic limit. X a ( τ , σ + ℓ ) = X a ( τ , σ ) + 2 π √ α ′ L a ; L a = D X i =1 n i e ai ; n i ∈ Z (3.5)where, X a ( σ, τ ) = e ai X i ( σ, τ ) . (3.6)The e i ≡ { e ai } can be interpreted to be D linearly independent vectors which generate a D dimensional lattice Λ D . The L ≡ { L a } in (3.5) are lattice vectors, i.e., L ∈ Λ D . The compactifiedtorus can now be viewed as the quotient of R D by π Λ D [22], i.e.: T D = R D π Λ D (3.7)We can also define a dual lattice by introducing vectors e ∗ i = { e ∗ ia } via : e ∗ ib e ai = δ ab ; e ai e ∗ ja = δ ji ; D X a =1 e ∗ ja e ∗ ia = g ij (3.8)We shall be mostly concerned with the zero modes of the compactified fields which correspond tothe massless fields in the lower dimension. For the metric, we express the general compactificationansatz as G MN = g µν + g kl A kµ A lν A µj A iν g ij ! ; G MN = g µν − A µj − A iν g ij + g ρσ A iρ A jσ ! (3.9)where we are assuming that all the quantities depend only on the non compact directions X µ ofthe space-time (since we are only interested in the zero-modes). Here A iµ are vector fields in thelower dimension. The g µν is the non compact space-time metric and g ij are the scalars in the lowerdimensional theory. Their expectation values can be related to the moduli of the compatified torus T D .The background values of g µν and g ij will be used to lower and raise the non compact andinternal indices respectively.Similarly, for the Kalb-Ramond field, we have B MN = B µν B µj B iν B ij ! , B µi = − B iµ (3.10)The B µi denote set of another vector fields in the lower dimension. B µν is an antisymmetric 2ndrank tensor field in the lower dimension. It will be taken to be vanishing in most of our discussion. The identity (3.4) also gives e ∗ jc g ij e ∗ id = P Da =1 e ai e ∗ id δ ab e bj e ∗ jc = δ dc , showing that the D-vectors e ∗ a ≡ ( e ∗ ia ) form an orthonormal basis. B ij denote other set of scalar fields in the lower dimension. Again, the expectation values of B ij can be related to the moduli of the compactified manifold T D .The infinitesimal coordinate transformation along the compact directions X i → X i + ξ i , where ξ depends only on the non-compact coordinates, changes the vector field A µi as δA µi ( X µ ) = g ij ∂ µ ξ i = ∂ µ ξ j ⇔ δA µa = ∂ µ ξ a ; A µa = e ∗ ia A µi (3.11)This is a U (1) gauge transformation acting on the vector field A µa . For our purposes, it will beenough to take the field strength of this gauge field to be a constant , in which case, we can write A µa = − F µν ; a X ν . (3.12)Similarly, the gauge transformation for the Kalb Ramond field is given by δB MN = ∂ M Λ N − ∂ N Λ M (3.13)Specializing this to a gauge parameter Λ i depending only on the non compact coordinates gives δB µi = ∂ µ Λ i (3.14)This is also a U (1) gauge transformation for the gauge field B µi . As above, we shall take the fieldstrength of this gauge field also to be a constant.As mentioned above, the metric g ij will be taken to have the constant expectation value givenby (3.4). For the other fields, we shall assume the following background values g µν = η µν ; B µν = 0 ; B ij = const (3.15)with η µν the Minkowski (mostly plus) metric for the non compact directions. T D The sigma model action in 26 dimensions in the presence of the space-time metric G MN ( X ) andan antisymmetric tensor B MN ( X ) is given by [23] S = − πα ′ Z d σ (cid:2) G MN ( X ) η αβ ∂ α X M ∂ β X N + B MN ( X ) ǫ αβ ∂ α X M ∂ β X N i (3.16)where, the space-time indices M, N run over , . . . d − with d = 26 . The world-sheet metric isgiven by η αβ = ( − , and our convention for the Levi civita tensor is ǫ = − ǫ = 1 . Also, σ α ≡ ( τ, σ ) .The canonically conjugate momenta of X M is given by Π M = δLδ∂ X M = 12 πα ′ h G MN ∂ X N − B MN ∂ σ X N i (3.17) If the field strength is not constant, there will be α ′ dependent corrections in (3.12) [19]. However, thesecorrections do not contribute to the gyromagnetic ratio since the terms with higher orders in α ′ correspond to thehigher derivative terms. H = Z ℓ dσ h Π M ∂ X M − L i = 12 Z ℓ dσ (cid:20) (2 πα ′ )Π M G MN Π N + 2 B MN G MP Π P ∂ σ X N + 12 πα ′ ∂ σ X M (cid:16) G MN + G P Q B P M B QN (cid:17) ∂ σ X N (cid:21) (3.18)For analysing the gyromagnetic couplings of the massive string states with respect to the gaugefields A iµ and B iµ , we shall need the terms linear in the gauge fields in the Hamiltonian describingthe interaction between these gauge fields and the world-sheet fields X i and X µ . This is easilyobtained, for the background defined by (3.15), by substituting the compactification ansatz (3.9)and (3.10) in the expression (3.18) H I = − πα ′ Z ℓ dσ (cid:20) A µi (cid:18) (2 πα ′ ) Π µ Π i + (2 πα ′ ) B ij Π µ ∂ σ X j − g µν g ij ∂ σ X ν ∂ σ X j (cid:19) + B µi (cid:16) − (2 πα ′ )Π µ ∂ σ X i + (2 πα ′ )Π i ∂ σ X µ + B jk g ki ∂ σ X µ ∂ σ X j (cid:17)(cid:21) (3.19)where, Π µ ≡ P µ + 12 πα ′ h A iµ ∂ X i − B µi ∂ σ X i − B µν ∂ σ X ν i + O ( A , B , AB ) (3.20) Π i ≡ P i + 12 πα ′ h A iµ ∂ X µ + B µi ∂ σ X µ i + O ( A , B , AB ) (3.21)with P µ = 12 πα ′ g µν ∂ X ν , P i = 12 πα ′ h g ij ∂ X j − B ij ∂ σ X j i . (3.22)In the Hamiltonian (3.19), being interested in the terms which are linear in the gauge fields, we canreplace Π µ,i with P µ,i . Furthermore by explicitly evaluating these quantities on the solution of theequations of motion given in equation (A.7), we find that all the dependence on the moduli B ij disappears and we can express (3.19) as H I = H AI + H BI (3.23)where, H AI = 12 πα ′ F Aµρ ; i Z ℓ dσ X ρ (cid:16) ∂ + X µ ∂ − X i + ∂ − X µ ∂ + X i (cid:17) H BI = 12 πα ′ F Bµρ ; i Z ℓ dσ X ρ (cid:16) ∂ + X µ ∂ − X i − ∂ − X µ ∂ + X i (cid:17) (3.24)12he expectation value for H AI which describes the interaction of the gauge field A µi with theworld-sheet fields is computed to be h φ |H AI | φ i = − m F Aµν ; i D φ (cid:12)(cid:12)(cid:12) L µν (cid:0) p iR + p iL (cid:1)| {z } Q i + p iR S µνL + p iL S µνR (cid:12)(cid:12)(cid:12) φ E (3.25)The p L and p R in the above expression denote the left and right momenta associated with stringcompactification and are defined in equation (A.10). The L µν denotes the orbital angular momen-tum and S µν = S µνL + S µνR denotes the spin angular momentum with S µνR = − i ∞ X n =1 n (cid:0) α µ − n α νn − α ν − n α µn (cid:1) , S µνL = − i ∞ X n =1 n (cid:0) ˜ α µ − n ˜ α νn − ˜ α ν − n ˜ α µn (cid:1) (3.26)In the following we shall identify these two operators with the names “right” and “left” spinoperators. In equation (3.25), the Kaluza-Klein charges of the massive higher spin states are givenby [19] Q a = 12 πα ′ Z ℓ dσ∂ τ X a = e ∗ ib ( p R + p L ) i δ ab (3.27)In a similar manner, for the gauge field B µi , we find h φ |H BI | φ i = − m F Bµν ; i h φ | L µν (cid:0) p iR − p iL (cid:1)| {z } Q i − S µνR p iL + S µνL p iR | φ i (3.28)with the winding charge of the U (1) field given by [19]: Q a = 12 πα ′ Z ℓ dσ∂ σ X i e ai = ( p L − p R ) i e ∗ ib δ ab (3.29)We notice that higher spin states with p R = p L or p R = − p L have null charges Q a or Q a respectively.From equations (3.25) and (3.28), we see that the gyromagnetic ratios of these particles are g = 1 with respect to one gauge field. Note also that when this happens the same massive field isuncharged with respect to the other gauge field being only coupled non-minimally to it with acoupling proportional to the difference of the left and right spin operators. Further commentsabout these special cases will be given in Section 5.It is worth noting that the expression (3.28) can be obtained from equation (3.25) by exchangingthe two gauge fields and changing the sign of the compact momentum p iL . The two charges, instead,are transformed into each other as ( p L , p R ) ↔ ( p L , − p R ) . In the case B ij = 0 , this transformationis the D -dimensional analogous of the R → /R -duality of the string theory compactified on acircle of radius R [22]. In Ref.[17] the KK-charge is taken equal to the compact momentum p z = n/R . In Eq. (3.27), when p R = p L ,the charge is ( p R + p L ) i e ∗ ia = n i √ α ′ e ∗ ia . The choice e ∗ ia = δ ia √ α ′ R a and e ai = δ ai R a √ α ′ matches the two charges and gives X a ≡ X a + 2 πR a n a which is the identification of the compact coordinates used in the above reference. .3 Type II and Heterotic strings on T D In this subsection, we consider the closed superstring theories and generalize the expression of theinteracting Hamiltonian of the bosonic string theory obtained in the previous subsection to bosonicand fermionic states of the superstring theories. We start by considering the type II string theorycompactified on the torus T D with D = 9 − d . After this, we shall extend this analysis to theHeterotic theory. Our approach will be exactly the same as in the case of the bosonic string theory.Thus, the starting point is the action of the (1 , supersymmetric sigma model in 10 dimensionsin a generic non constant background [24, 25, 26] (see appendix A for details) S = 14 πα ′ Z d σ (cid:20) G MN ∂ + X M ∂ − X N + 4 B MN ∂ + X M ∂ − X N + 2 iG MN ψ M + ˜ ∇ − ψ N + + 2 iG MN ψ N − ˜ ∇ + ψ M − + 12 ˜ R MNP Q ψ M + ψ N + ψ P − ψ Q − (cid:21) (3.30)where, the covariant derivatives ˜ ∇ ± are defined by ˜ ∇ ± ψ M ∓ = ∂ ± ψ M ∓ + ˜Γ M ± P Q ψ P ∓ ∂ ± X Q , ˜Γ M ± P Q = Γ
MP Q ± H MP Q (3.31)The ˜Γ P ± MN are the connections with a totally antisymmetric torsion. The ˜ R MNP Q are given by ˜ R MNP Q = R MNP Q + 12 ∇ P H MNQ − ∇ Q H MNP + 14 H MRP H RQN − H MRQ H RP N (3.32)The conjugate momenta for X M are computed to be Π M = 12 πα ′ h G MN ∂ X N − B MN ∂ σ X N − S M i (3.33)where, S M = − i G P Q (cid:20) ˜Γ P − NM ψ Q + ψ N + + ˜Γ P + NM ψ Q − ψ N − (cid:21) (3.34)The conjugate momenta corresponding to ψ M ± are given by τ M ± = δLδ∂ ψ M ± = i πα ′ G MN ψ N ± (3.35)The Hamiltonian is computed to be H = 14 πα ′ Z ℓ dσ " (2 πα ′ ) G MN Π M Π N + 2(2 πα ′ ) G MN Π M S N + 2(2 πα ′ ) G MN Π M B NP ∂ σ X P +2 G MN B MP ∂ σ X P S N + G MN B MP B NQ ∂ σ X P ∂ σ X Q + G MN S M S N + G MN ∂ σ X M ∂ σ X N − iG MN (cid:16) ψ N − ∂ σ ψ M − − ψ N + ∂ σ ψ M + (cid:17) +2 T P ∂ σ X P −
12 ˜ R SMLP ψ S + ψ M + ψ L − ψ P − (3.36)14here, we have defined T P ≡ i G P Q (cid:16) ˜Γ P − NM ψ Q + ψ N + − ˜Γ P + NM ψ Q − ψ N − (cid:17) (3.37)We can now compute the part of Hamiltonian which describes the interaction between the stringstates and the gauge fields resulting from the compactification on T D . For convenience, we organisethem in two kind of terms: one in which strings interact with the external gauge field and the other inwhich they interact with their field strength. The terms describing the interaction of the world-sheetfields with only one gauge field are given by H = 12 πα ′ Z ℓ dσ (cid:20) A µi n − (2 πα ′ ) Π µ Π i − (2 πα ′ )Π µ B i j ∂ σ X j + ∂ σ X i ∂ σ X µ + i (cid:16) ψ µ + ∂ σ ψ i + − ψ µ − ∂ σ ψ i − + ψ i + ∂ σ ψ µ + − ψ i − ∂ σ ψ µ − (cid:17)o + B µi n (2 πα ′ )Π µ ∂ σ X i + (2 πα ′ )Π i ∂ σ X µ + g ij B jk ∂ σ X k ∂ σ X µ o(cid:21) (3.38)and the terms describing the interaction with the field strength are given by H = − i Z ℓ dσ (cid:20) F Aµν ; i n Π µ Ψ νi + − Π i Ψ µν + − πα ′ (cid:16) B i j ∂ σ X j Ψ µν + + ∂ σ X µ Ψ νi − − ∂ σ X i Ψ µν − (cid:17)o + F Bµν ; i n − µ Ψ νi − − Π i Ψ µν − + 12 πα ′ (cid:16) − B i j ∂ σ X j Ψ µν − + 2 ∂ σ X µ Ψ νi + + ∂ σ X i Ψ µν + (cid:17)o(cid:21) (3.39)where, we defined Ψ MN ± = ψ M + ψ N + ± ψ M − ψ N − .The expectation value of the interacting Hamiltonian between two generic string states is givenby h φ |H I | φ i = h φ |H | φ i + h φ |H | φ i (3.40)We have divided equations (3.38) and (3.39) by the factor introduced in (3.1) to have the canonicalnormalization. The state | φ i now also includes the fermionic oscillators along with the bosonic ones.The calculation of the first term is exactly identical to the calculation of the bosonic case becausethe second line of (3.38) gives zero contribution when evaluated on the external states. This canbe seen by inserting the mode expansion for the fields and noting that these terms change the levelof the state and hence the inner product becomes zero. The remaining terms in (3.38) are exactlyidentical to the Bosonic Hamiltonian.Thus, we only need to focus on the second term of the above expression which involves H .As in the bosonic case, we can again replace the conjugate momenta Π m and Π µ by P m and P µ respectively, given in (3.22), upto the linear order in the fields. Moreover, the terms proportional to ψ µ ± ψ m ± give vanishing contribution since they change the level of the states and the inner product15anishes due to orthogonality property. The expression can then be expressed in the form h φ |H | φ i = i πα ′ F Aµν ; i Z ℓ dσ (cid:10) φ (cid:12)(cid:12)(cid:8) ∂ − X i ψ µ + ψ ν + + ∂ + X i ψ µ − ψ ν − (cid:9)(cid:12)(cid:12) φ (cid:11) + i πα ′ F Bµν ; i Z ℓ dσ (cid:10) φ (cid:12)(cid:12)(cid:8) ∂ − X i ψ µ + ψ ν + − ∂ + X i ψ µ − ψ ν − (cid:9)(cid:12)(cid:12) φ (cid:11) (3.41)We now consider the case B µi = 0 . Using the mode expansions given earlier and performing the σ -integration, we find h φ |H A | φ i = i m F Aµν ; m X r ∈ Z + a D φ (cid:12)(cid:12)(cid:12)n p mR ¯ ψ µr ¯ ψ ν − r + p mL ψ µr ψ ν − r o(cid:12)(cid:12)(cid:12) φ E = − m F A aµν D φ (cid:12)(cid:12)(cid:12)n ( p R + p L ) a ( K µνL + K µνR ) + ( p L − p R ) a ( K µνR − K µνL ) o(cid:12)(cid:12)(cid:12) φ E (3.42)where, K µνL ≡ − i ψ µ , ¯ ψ ν ] δ a, − i X r ∈ N + a ( ¯ ψ µ − r ¯ ψ νr − ¯ ψ ν − r ¯ ψ µr ) (cid:17) K µνR ≡ − i ψ µ , ψ ν ] δ a, − i X r ∈ N + a (cid:16) ψ µ − r ψ νr − ψ ν − r ψ µr (cid:17) (3.43)are the contribution to the angular momentum from the ψ M ± fields. This expression, when addedto hE |H A |E i changes the bosonic result by the replacement of S R,L given in Eq.(3.26) and nowdenoted with S BR,L with S R,L = S BR,L + K R,L , the spin operators in superstring theory, giving h φ |H AI | φ i = − m ( F A ) aµν Q a (cid:10) φ (cid:12)(cid:12) ( L µν + S µν ) + Q a Q a ( S µνR − S µνL ) (cid:12)(cid:12) φ (cid:11) (3.44)where the charges are defined in equations (3.27) and (3.29).For the case of non zero B µi , the calculation proceeds in the similar way. By looking at (3.41),we find that the only difference in the calculation involving B µi as compared to A µi is in the signin front the second term containing ∂ + X m . This corresponds to the replacement of p L with − p L and therefore in the exchange of Q a with Q a in the final expression, giving h φ |H B | φ i = − m F A aµν D φ (cid:12)(cid:12)(cid:12)n ( p R − p L ) a ( K µνL + K µνR ) + ( p L + p R ) a ( K µνL − K µνR ) o(cid:12)(cid:12)(cid:12) φ E (3.45)Again, when we add the contribution from H , which is same as given in equation (3.28), it changesthe bosonic result by the replacement ( S BR,L ) µν → S µνR,L + K µνR,L , the spin operators in supersymmetrictheory. Finally, we turn to the SO (32) and E × E Heterotic theories compactified on the torus T D . Again, we shall compute the Hamiltonian, giving the interaction between string world sheetfields and the gauge fields. The starting point will be the heterotic sigma model in ten dimensionsin the presence of the background fields G MN , B MN and the gauge field ( A M ) BC . The gauge16roup in the E × E or SO (32) models could be equivalently represented by fermionic or bosonicformulations. In the following, we follow the former approach and start from an action containing Majorana-Weyl fermions λ A − coupled to the background fields. The indices of λ A − are loweredand raised by the metric g AB . The sigma model action turns out to be [24, 26, 27, 28] S = 14 πα ′ Z d σ h G MN ∂ + X M ∂ − X N + 4 B MN ∂ + X M ∂ − X N + 2 iG MN ψ M + ˜ ∇ − ψ N + +2 ig AB λ A − ˆ ∇ + λ B − + 12 F MN ; CD ψ M + ψ N + λ C − λ D − i (3.46)where the ψ + are left-moving fermions and ˜ ∇ − ψ N + is defined in the same way as in the type II case(see equation (A.32)) and ˆ ∇ + λ B − = ∂ + λ B − + ( ˆ A M ) BC λ C − ∂ + X M , ( ˆ A M ) BC = ( A M ) BC + 12 g BD ∂ M g DC (3.47)The field strength for the gauge field ( ˆ A M ) BC is defined as F MN ; CD = ∂ M ( ˆ A N ) CD − ∂ N ( ˆ A M ) CD + ( ˆ A M ) CB ( ˆ A N ) BD − ( ˆ A N ) CB ( ˆ A M ) BD (3.48)The conjugate momenta for X M are given by Π M = 12 πα ′ h G MN ∂ X N − B MN ∂ σ X N − S M i (3.49)where, S M is now given by S M = − i G P Q ˜Γ P − NM ψ Q + ψ N + − i g AB ( ˆ A M ) BC λ A − λ C − (3.50)The conjugate momenta of ψ M + is same as in the case of type II superstrings. For the λ A − , we have Π A − = δLδ∂ λ A − = i πα ′ g AB λ B − (3.51)The Hamiltonian is computed to be H = 14 πα ′ Z ℓ dσ " (2 πα ′ ) G MN Π M Π N + 2(2 πα ′ ) G MN Π M S N + 2(2 πα ′ ) G MN Π M B NP ∂ σ X P +2 G MN B MP ∂ σ X P S N + G MN B MP B NQ ∂ σ X P ∂ σ X Q + G MN S M S N + G MN ∂ σ X M ∂ σ X N + iG MN ψ N + ∂ σ ψ M + + 2 T P ∂ σ X P − ig AB λ A − ∂ σ λ B − − F MN ; AB ψ M + ψ N + λ A − λ B − (3.52)where, T M ≡ i G P Q ˜Γ P − NM ψ Q + ψ N + − i g AB ( ˆ A M ) BC λ A − λ C − (3.53)17or doing calculations, it is useful to note that the terms in the Hamiltonian (3.52) which do notinvolve λ A − are same as the corresponding terms in the type II Hamiltonian (3.36).We can now simplify the interaction terms for the gauge fields A µm , B µm and ( ˆ A M ) BC . Weagain organise them in two kind of terms. One in which strings interact with one external gaugefield and the other in which they interact with their field strength. They are given by H = 12 πα ′ Z ℓ dσ (cid:20) A µi n − (2 πα ′ ) Π µ Π i − (2 πα ′ )Π µ B in ∂ σ X n + ∂ σ X i ∂ σ X µ + i (cid:16) ψ µ + ∂ σ ψ i + + ψ i + ∂ σ ψ µ + (cid:17)o + ( ˆ A µ ) CD n − i πα ′ )Π µ λ CD − − i ∂ σ X µ λ CD − o + B µi n (2 πα ′ )Π µ ∂ σ X i + (2 πα ′ )Π i ∂ σ X µ + g ij B jk ∂ σ X k ∂ σ X µ o(cid:21) (3.54)and, H = − i Z ℓ dσ (cid:20) F Aµν ; i n Π µ ψ νi + − Π i ψ µν + − πα ′ (cid:16) B i j ∂ σ X j ψ µν + + ∂ σ X µ ψ νi + − ∂ σ X i ψ µν + (cid:17)o + F Bµν ; i n − µ ψ νi + + Π m ψ µν + + 12 πα ′ (cid:16) B i j ∂ σ X j ψ µν + + 2 ∂ σ X µ ψ νi + − ∂ σ X i ψ µν + (cid:17)o − i πα ′ F µν ; CD ψ µν + λ CD − (cid:21) (3.55)We only focus on the gauge fields A µi and B µi and work with the rescaled Hamiltonian introducedin equation (3.1). The expectation value of H between the two generic string states is againexactly identical to the calculation of the bosonic case since the term involving the ψ µ + and λ A − fields do not contribute. Thus, we need only to focus on the expectation value of H . We firstconsider the case B µi = 0 . In this case, by looking at the expressions, we find that the calculationwill be exactly identical to the type II case except that we need to drop the terms involving ψ M − .This gives h φ |H A | φ i = iℓ πα ′ ) m F Aµν ; i Z ℓ dσ (cid:10) φ (cid:12)(cid:12) ∂ − X i ψ µ + ψ ν + (cid:12)(cid:12) φ (cid:11) = i m F Aµν ; i X r ∈ Z + a (cid:10) φ (cid:12)(cid:12) p iR ¯ ψ µr ¯ ψ ν − r (cid:12)(cid:12) φ (cid:11) = − m F Aµν ; i (cid:10) φ (cid:12)(cid:12) p iR K µνL (cid:12)(cid:12) φ (cid:11) (3.56)where K L is defined in equation (3.43).Similarly, the calculation for non zero B µi proceeds in the same way as type II case except thatwe need to forget about the terms involving ψ µ − . This gives h φ |H B | φ i = − m F Bµν ; i (cid:10) φ (cid:12)(cid:12) p iR K µνL (cid:12)(cid:12) φ (cid:11) (3.57)One point to note about the above results is that λ A − are Lorentz singlets. Hence, they do notcontribute to the spin angular momentum which is reflected in the above expressions.18inally by combining equations (3.56) and (3.57) with the contribution coming from H as givenin equations (3.25) and (3.28) respectively, we find that these expressions, i.e. (3.25) and (3.28),are still valid in Heterotic sigma model but with the spin operators replaced by the appropriateexpressions. Gyromagnetic factors can be extracted also from string 3pt amplitudes. It turns out that the valueof the gyromagnetic factor is entirely encoded within the graviton vertex operator which upondimensional reduction produces a contribution to the effective string action precisely of the type(2.10).The fact that gyromagnetic factors are entirely encoded in the graviton vertex operator showthat consistent electromagnetic couplings are related to consistent minimal couplings to gravitybefore compactification. Uniqueness of the minimal coupling to gravity than translates into highlyconstrained gyromagnetic factors.Therefore, in this section we compute the gyromagnetic ratios of arbitrary higher spin states ofthe bosonic and superstring theories by computing three-point functions of massive high spin statesand U (1) gauge fields emerging from the compactification procedure. The starting point is eitherthe bosonic, superstring or heterotic string theories in the critical dimensions, d = 26 or d = 10 ,compactified on the torus T D ≡ R D / π Λ D , with Λ D the lattice introduced in Sec. § The compactification generates U (1) -gauge fields and we will focus on those coming from the d -dimensional gravitons or the Kalb-Ramonds fields with one index extended along the compactdirections and the other one non compact. In bosonic string theory the -dimensional masslessfields, graviton, dilaton and Kalb-Ramond, are described by the vertex operator: V g ( z, ¯ z ) = ε MN ∂ z X MR ( z ) ∂ ¯ z X NL (¯ z ) e q α ′ i p M ( X MR ( z )+ X ML (¯ z )) , (4.1)with z = e i πl ( τ − σ ) and ¯ z = e i πl ( τ + σ ) . To make explicit the factorization properties of the threepoint amplitudes in string theory, we decompose, as usual, the polarization of the massless state ε MN = ǫ M × ¯ ǫ N and we define the left and right vertices as follows: V g ( z ) = ǫ M ∂ z X MR ( z ) e q α ′ i p M X MR ( z ) , V g (¯ z ) = ¯ ǫ M ∂ z X ML ( z ) e q α ′ i p M X ML (¯ z ) . (4.2)In the compactification procedure we require that all the components of the d -dimensionalmassless state, remain massless at d − D -dimensions. This is achieved by keeping the momentumcarried by the vertex different from zero only along the non-compact directions, i.e. p M = ( p µ , .19he massive high-spin states, instead, can carry momenta in both compact and non-compactdirections : p MR,L ≡ (cid:18) p µ , p aR,L (cid:19) ; p L,R ; a = 12 √ α ′ (cid:2) n i + B ij m j ± g ij m j (cid:3) e ∗ ia . (4.3)In equation (4.1) one can now replace the expression of the field X ( z ) given by X R ( z ) = ˆ q − iα ln z + i X n =0 α n n z − n , (4.4)with α defined in Eq. (A.12) and a similar expression for X L (¯ z ) . Here ˆ q represents the coordinateof the string center of mass which acts on string states.The amplitudes that we want to compute involves one gauge field described by the vertex (4.1)and two identical high-spin states, of the level N = n R + n L . Here, n L,R are the eigenvalues of theleft and right-number operators. The 3-point function can then be expressed in the factorized form A = 2 κ d (cid:18) α ′ (cid:19) h n R , p R | V g ( z = 1) | n R , p R i ∧ h n L , p L | V g (¯ z = 1) | n L , p L i . (4.5)Here, we have already used the SL (2 , C ) invariance of the world-sheet CFT to fix the Koba-Nielsenvariables of the massive high-spin vertices to z = 0 , ∞ , while that of the massless vertex is at thepoint z = ¯ z = 1 .From the above expression (4.5) we shall now determine the gyromagnetic ratio of the corre-sponding high-spin states. To this end it will be sufficient to consider an α ′ expansion of the vertexoperator. The leading orders in α ′ are indeed sufficient to read off the couplings (2.1).Working at the level of the holomorphic part of the above correlator one then gets: h n R , p R | V g ( z = 1) | n R , p R i = ǫ (2) M D n R , p R (cid:12)(cid:12)(cid:12) : ∂ z X MR (1) (cid:16) − r α ′ X n =0 p · α n n (cid:17) : (cid:12)(cid:12)(cid:12) n R , p R + p E + O ( √ α ′ ) . (4.6)Carrying out the algebra substituting the explicit expressions for ∂ z X MR (1) one the arrives to: h n R , p R | V g ( z = 1) | n R , p R i = − i r α ′ D n R , p R (cid:12)(cid:12)(cid:12)h ǫ (2) M p M R − i p M ǫ (2) N − p N ǫ (2) M ) | {z } F MN ( p ) / ˆ S MNR i(cid:12)(cid:12)(cid:12) n R , p R + p E + O ( √ α ′ ) , (4.7)where we have introduced the right spin operator defined in Eq. (3.26) and replaced the momentumoperator with its eigenvalue. We have also imposed the on-shell condition ǫ (2) · p = 0 . Similar In the case of the graviton vertex, one could introduce the same momentum notation adopted for the higherspin states, by defining p R ; L ≡ ( p , . In this case the exponential factor of the graviton vertex has to be writtenin the form e i √ α ( p R · X R + p L · X L ) . V ( ǫ ¯ ǫ )ΦΦ = − α ′ D Φ (cid:12)(cid:12)(cid:12) (cid:20) ǫ · p R − F MN ( p ) ˆ S MNR (cid:21) (cid:20) ¯ ǫ · p L −
12 ¯ F ¯ M ¯ N ( p ) ˆ S ¯ M ¯ NL (cid:21) (cid:12)(cid:12)(cid:12) Φ E . (4.8)The ( d − D ) -dimensional U (1) -gauge fields are obtained by considering polarization tensors withmixed space time indices, one non-compact and the other compact. There exist two possibilities: A µa ≡ (cid:16) ε µ ¯ ε a + ε a ¯ ε µ (cid:17) , (4.9) B µa ≡ (cid:16) ε µ ¯ ε a − ε a ¯ ε µ (cid:17) . (4.10)with µ = 0 . . . d − D − and a = 1 , . . . D . In terms of these quantities eq.(4.8) becomes: V ( A,B )ΦΦ = − α ′ D Φ (cid:12)(cid:12)(cid:12) ( p aL + p aR ) | {z } Q a A a · p − F Aµν ; a ( p a L S µνR + p a R S µνL )+ ( p aL − p aR ) | {z } Q a B a · p − F Bµν ; a ( p a L S µνR − p a R S µνL ) (cid:12)(cid:12)(cid:12) Φ E , (4.11)where we have performed the replacements: ǫ µ ¯ ǫ a → A µa + B µa , ¯ ǫ µ ǫ a → A µa − B µa , (4.12)and where we have focused on contribution to the minimal and gyromagnetic couplings so that F Aµν ; a , F Bµν ; a are the field strengths in the momentum space of the gauge fields defined in Eq.(4.10),i.e.: F Aµν ; a = i ( p µ A νa − p ν A µa ) , F Bµν ; a = i ( p µ B νa − p ν B µa ) . (4.13)From eq. (4.11) one can read off the charges from the coefficient of the minimal coupling for A and B respectively while the gyromagnetic factor is expressed in terms of the right and left spin-operators therefore producing a particular combination of (2.10) with appropriate coefficients α ( j ) .This will require in particular to relate (depending on the representation considered) S R and S L tothe canonical spin operators S j . The bosonic case case can be easily extended to closed superstring theory. The three-point ampli-tude to be computed in superstring is the same as in the case of the bosonic theory. The difference Note that since the momentum p is entirely non-compact the only possible non-vanishing reduction of F MN is F µν .
21s that the two external high-spin states are now taken in superghost picture ( − , − while themassless states is in the zero picture. The amplitude where the SL (2 , C ) invariance has been fixedby choosing z = ∞ , z = 1 and z = 0 is: A = 2 κ d (cid:16) α ′ (cid:17) ǫ (2) M h n R , p R | : h ∂ z X MR − r α ′ ip ,R · ψ − (1) ψ M − (1) i e i q α ′ p · X R (1) : | n R , p R i ∧ L-sect.(4.14)The three point amplitude is now evaluated along the same lines of the bosonic calculation focusingon each open string sector. The expansion of the exponential gives: A = 2 κ d ( − i ) ǫ (2) M h n R , p R | : ˆ p M − p N h ∞ X n =1 α M − n α Nn − α N − n α Mn n + δ a ;0
12 [ ψ M , ψ N ]+ X r ∈ Z + a ( ψ M − r + a ψ Nr + a − ψ N − r + a ψ Mr + a ) i : | n R , p R + p i ∧ L-sect. + O ( √ α ′ )= 2 κ d ( − i ) ǫ (2) M h n R , p R | h p M − ip N ˆ S MNR i | n R , p R + p i ∧ L-sect. + O ( √ α ′ ) , (4.15)with ˆ S MNR = − i ∞ X n =1 α M − n α Nn − α N − n α Mn n − iδ a ;0
12 [ ψ M , ψ N ] − i X r =1+ a ( ψ M − r + a ψ Nr + a − ψ N − r + a ψ Mr + a ) , (4.16)where a = 0 , / in the R and NS sector respectively.The right correlator, given in eq. (4.15), formally coincides with that written in eq. (4.7), the onlydifference is in the explicit form of the spin-operators which in the superstring case depends on alsofermionic oscillators entering the massive string vertices. Therefore, the amplitude in superstring,to the leading order in the string slope, is formally identical with the one computed in bosonic stringtheory.Finally, the vertex operator of the massless state in Heterotic string, in one sector is equal tothe one of the superstring while in the other sector coincides with the one of bosonic theory. Thethree-point amplitude, in Heterotic string with a massless vertex and two massive high-spin states,to leading order in α ′ , can therefore be obtained by combining the left and right correlators of thebosonic and superstring theories, respectively. This amplitude turns out to be formally identicalto the corresponding expressions found in the bosonic and superstring models, the differencesare, again, in the explicit realisation of the spin-operators defined on each sector. Consequently,the gyromagnetic factor obtained in the bosonic string case is still valid in these string models.These results, obtained from amplitude calculations, are in agreement with those obtained fromthe hamiltonian approach developed in the previous sections and confirm the universality of Eq.s(3.25) and (3.28) from which one can deduce the gyromagnetic ratios of higher-spin particles instring theory. 22n the following section we shall consider a few examples with the aim of reducing the combi-nation of S R and S L appearing in (4.11) to the form (2.10) and extract the explicit expressions forthe gyromagnetic factors α ( j ) . This section is devoted to some explicit examples which involve the states of first Regge trajectoryof closed string theory. We shall focus both on totally symmetric fields and hook-fields and extractthe corresponding gyromagnetic ratios from (4.11). p R = ± p L In this special case, where the left and right compact momenta are equal modulo a sign, correspondsto vanishing Kaluza Klein ( p L = − p R ) or Winding charges ( p L = p R ) defined in equations (3.27) and(3.29) respectively. In these cases, the expressions of the gyromagnetic ratios simplify considerably.This happens because, in these cases, the relevant interactions depend on the combination S = S L + S R which allows us to read off the gyromagnetic ratio for arbitrary elements of the spectrumregardless of the Young Tableaux representation. The gyromagnetic factor in this case is g = 1 .This generalises the results previously obtained in different contexts for massive spin two particles[16, 17, 18]. It is interesting to compare the value g = 1 for the gyromagnetic ratio obtained in the previoussubsection with the value g = 2 which is obtained by requiring consistent HS electromagneticinteractions in constant curvature backgrounds [11]. In this section, we shall argue that the universalnature of g = 1 which is seen in the context of string compactifications is a direct consequenceof the uniqueness of gravitational minimal coupling. This clarifies the universality of g = 1 in thecontext of field theory compactifications. Such universality will be lost when considering windingstates and more general values of the gyromagnetic factor are possible in these cases as we shallsee. Note that the Kalb-Ramond field can couple to the lowest order in derivatives with two derivative interactions.In the case of totally symmetric fields one has for instance: V (1) Bφφ = B MN (cid:16) ∂ M φ ⋆R ( s ) (cid:17) (cid:16) ∂ N φ R ( s ) (cid:17) , (5.1) V (2) Bφφ = H MNL (cid:16) ∂ M φ ⋆NR ( s − (cid:17) (cid:16) φ LR ( s − (cid:17) . (5.2)While the first coupling gives minimal coupling upon compactification the second contributes to the gyromagneticratio. In String theory this coupling is fixed by the double-copy structure of the vertex operator. This is required byT-duality relating the coupling of the Kalb-Ramond field to the coupling of the graviton. Similar considerations canbe made for more general representations including fermions. We thank Ashoke Sen for discussions on this point.
23o do so it is useful to analyse which couplings would give a contribution to the gyromagneticfactor upon dimensional reduction. We can restrict to two derivative couplings because these arethe only couplings which upon dimensional reduction will produce couplings with a single derivative.Using the classification of cubic couplings obtained in [29, 30, 31] we then get three possible twoderivative gravitational couplings: ( u · u ) s − h ( u · p ) ( u · u ) + β s u · p ( u · p u · u + u · p u · u )+ γ s ( u · p u · u + u · p u · u ) i . (5.3)Their dimensional reduction then gives (focusing on the EM coupling): nR ( u · u ) s − h ( u · p )( u · u ) + β s ( u · p u · u + u · p u · u ) i , (5.4)from which one recovers g = β s s . (5.5)The first observation is that it does not depend on γ s which is the coefficient of the non-minimalcoupling proportional to the curvature R . The coupling proportional to γ s does not deform theabelian gauge symmetries and is therefore arbitrary in principle. The second observation is thatthe induced gyromagnetic factor is given by (5.5) and one might think that β s could be arbitrary!However the corresponding coupling deforms the gauge transformations of the spin- s field! Itmust therefore be fixed by the requirement that the induced gauge transformations match the Liederivative if the HS field are consistently coupled to gravity.Evaluating the deformation of the gauge transformations using eq. (3.7) of [32] we obtain: s ! u µ · · · u µ s ( δ ξ Φ) µ ...µ s = 1 s ! ( u · u ) s − h u · p ( u · u ) + β ( − u · p u · u + u · p u · u ) i = 1 s ! ( u · u ) s − h u · p ( u · u ) + β u · p u · u − β u µ u ν ( p ,µ u ,ν + p ,ν u ,µ ) i (5.6)where the last term that we have set apart can be reabsorbed by a trivial field redefinition beingproportional to the symmetrized gradient of the gauge parameter. Starting instead from the Liederivative and considering the replacement (B.1) we get s ! u µ · · · u µ s L ξ φ µ ...µ s = 1 s ! u µ · · · u µ s (cid:2) ξ µ ∂ µ φ µ ··· µ s + s ( ∂ ( µ ξ ν ) φ ν | µ ··· µ s ) (cid:3) → s ! u µ · · · u µ s ( L ξ Φ) µ ··· µ s = 1 s ! ( u · u ) s − h u · p ( u · u ) + s u · p u · u i (5.7)24equiring that up to trivial redefinition the gauge transformation match among each other forcesthen β s = s . Therefore, we have shown that g = 1 is a consequence of the uniqueness ofgravitational minimal coupling since the minimal coupling is the only coupling which contributes tothe gyromagnetic factor upon reduction on the circle. The overall factor in the Lie derivative canbe used to fix the overall normalisation of the cubic gravitational coupling.It is also straightforward to extract γ s . We get γ s = s in all closed string theories although γ s cannot be fixed by requiring consistency of minimal coupling. It is interesting to note that with thischoice of γ s the 2-derivative gravitational coupling of all closed string theories takes the followingsimple form ( u · u ) s − h u · p u · u + s u · p u · u + u · p u · u ) i . (5.8)If one considers on the other hand the interaction of open string states with the graviton, oneobtains γ s = 0 . It would be interesting to understand if these are the only possible choices for γ s in consistent theories of gravity. The case of totally symmetric fields in the first Regge trajectory is the simplest. These are describedby Young Tableau diagrams having a single row. In this case, there exist a single gyromagneticratio so that both S R and S L must contribute to the same structure (2.5).To extract the gyromagnetic ratio, we can use the following identity D Φ (cid:12)(cid:12)(cid:12) xS µνL + yS µνR (cid:12)(cid:12)(cid:12) Φ E α = 1 s ( xℓ L + yℓ R ) (cid:10) Φ (cid:12)(cid:12) S µν (cid:12)(cid:12) Φ (cid:11) u , (5.9)which is derived in appendix D. This gives α = 1 s ( xℓ L + yℓ R ) , (5.10)in terms of right and left spins. The gyromagnetic ratio is then obtained by dividing with the chargeassociated to the corresponding gauge field g ( a ) A = 2 p aL + p aR p aL ℓ R + p aR ℓ L ℓ R + ℓ L , g ( a ) B = 2 p aL − p aR p aL ℓ R − p aR ℓ L ℓ R + ℓ L . (5.11)The above equation shows how for all states which satisfy level matching p L = p R , one recovers g ( a ) A,B = 1 . Note that for the gauge field B technically the charge goes to zero but one can still define the gyromagneticratio in the limit. It is however possible to obtain values of the gyromagnetic ratio different from one whenever levelmatching is not satisfied which happens for string states with non-trivial winding along the compact directions. .4 Mixed-Symmetry The mixed symmetry case is more complicated in general but it is the generic case within the stringspectrum when compactifications to d > are considered. We focus here on the example of tworow Young Tableaux which appear in the first Regge trajectory of the closed bosonic string.In this case, starting from the product of two totally symmetric representations of spin ℓ R and ℓ L with ℓ R ≥ ℓ L , associated to the first Regge trajectory of the open string, one has to project ontothe irreducible component associated to the tableaux { ℓ R + ℓ L − k, k } . To obtain the gyromagneticratio, we can make use of the identity D Φ (cid:12)(cid:12)(cid:12) xS µνL + yS µνR (cid:12)(cid:12)(cid:12) Φ E = (cid:10) Φ (cid:12)(cid:12) α S µν + a S µν (cid:12)(cid:12) Φ (cid:11) u , (5.12)where on the left hand side we have the closed string-correlator and on the right-hand side we usedthe inner-products among Young Tableaux as described in Appendix D. Considering the explicitprojection on the two-row Young Tableaux we obtain α = ( ℓ R − k ) x + ( ℓ L − k ) yℓ R + ℓ L − , α = ( ℓ R − k ) y + ( ℓ L − k ) xℓ R + ℓ L − . (5.13)We can then read off the gyromagnetic ratios for the gauge field A aµ g ( a )1 = 2 p aL + p aR ( ℓ R − k ) p aL + ( ℓ L − k ) p aR ℓ R + ℓ L − k , (5.14) g ( a )2 = 2 p aL + p aR ( ℓ R − k ) p aR + ( ℓ L − k ) p aL ℓ R + ℓ L − k , (5.15)as well as for the gauge field B aµ g ( a )1 = 2 p aL − p aR ( ℓ R − k ) p aL − ( ℓ L − k ) p aR ℓ R + ℓ L − k , (5.16) g ( a )2 = 2 p aL − p aR − ( ℓ R − k ) p aR + ( ℓ L − k ) p aL ℓ R + ℓ L − k , (5.17)Similar expressions follow from for any mixed-symmetry representation in subleading Regge trajec-tories. In this work, we considered Bosonic, Type II and Heterotic string theories compactified on a generic D dimensional torus in the presence of constant moduli. We focused on the interaction betweenthe U (1) gauge fields emerging from the dimensional reduction of the Graviton and Kalb-Ramondfields with the massive HS string states. The d -dimensional diffeomorphism invariance, broken by Below, the inner products are of order one in the permutation of indices. g = 1 and depend upon the spinand charges in the general case. For states with mixed symmetry properties described by a genericYoung Tableau, there are multiple gyromagnetic couplings, with one per row of the Young diagram.These gyromagnetic couplings were read by projecting onto the appropriate states as described inappendix D.Our results show how all values of the gyromagnetic factors in closed string theories turn out tobe a simple property of the structure of the graviton vertex operator. Interestingly we also point outhow to obtain values of g different from one by turning on winding charges. While our approachcan be considered top-down, starting from a consistent theory like string theory and deriving thevalue of the gyromagnetic factor, it would be very interesting to investigate this problem from abottom-up perspective by deriving how these values of the gyromagnetic factor are consistent withbasic principles. For field theory modes this is indeed possible, as one can easily see that g = 1 isimplied by minimal coupling with gravity. It is tempting to think that a similar story should holdalso for modes with non-trivial winding, which however cannot be studied in field theory.It would be interesting to shed some light on the universality of the expression giving thegyromagnetic ratios in other examples such as flux compactifications and string theories on orbifoldsbreaking partially or totally space-time supersymmetry. A similar analysis can also be performed inprinciple in curved backgrounds like AdS or dS where one can use the consistency of the boundarycorrelators to constraint the gyromagnetic factor also in the context of Inflation. Furthermore, itwould be interesting to study the connection of these results to fundamental properties like causalitydirectly at the level of the observables focusing e.g. to the AdS and dS cases where causality canbe mapped to concrete properties of the dual CFT correlators along similar lines as in [33]. Weleave this as well as other interesting related questions for future work. Acknowledgement:
We are thankful to Dario Francia, and Charlotte Sleight for useful dis-cussions and to Paolo Di Vechia, Rajesh Gopakumar, Massimo Porrati, Ashoke Sen and Kostas27kenderis for key comments on this draft. M.V. is also thankful to the organisers of 15th KavliAsian Winter School on Strings Particle and Cosmology in which some of the results of this workwere presented. The research of M.T. was partially supported by the program “Rita Levi Mon-talcini” of the MIUR (Minister for Instruction, University and Research) and the INFN initiativeSTEFI.
A Some details of the Hamiltonian Computation
A.1 Bosonic sigma model
The Bosonic sigma model in 26 dimensions in the presence of G MN and B MN is described by S = − πα ′ Z d σ (cid:2) G MN ( X ) η αβ ∂ α X M ∂ β X N + B MN ( X ) ǫ αβ ∂ α X M ∂ β X N i (A.1)This action is invariant under a general coordinate transformations and changes by a total derivativeunder the gauge transformation δB MN = ∂ M Λ N − ∂ N Λ M (A.2)The equation of motion of X M coming from the action (A.1) is ∂ α ∂ α X P + Γ PMN ∂ α X M ∂ α X N − H PMN ǫ αβ ∂ α X M ∂ β X N = 0 (A.3)with Γ PMN denoting the Christoffel symbols and H PMN = G P R H RMN ; H MNP = (cid:16) ∂ P B MN + ∂ N B P M + ∂ M B NP (cid:17) (A.4)The equation of motion (A.3) can be easily solved for the compactification described in section 3.1.For the background (3.15) and at the linear order in the field strengths, only the Γ µνm componentsof the Christoffel symbols contribute and are given by Γ µνi = 12 η µσ F Aνσi (A.5)Similarly, only the H µνi = F Bµνi components of the H MNP contribute. With these, the equations ofmotion for X i and X µ become ∂ α ∂ α X i − g ij F Bµν ; j ǫ αβ ∂ α X µ ∂ β X ν + O ( F A , F B , F A F B ) = 0 ∂ α ∂ α X µ − η µν F Aνσ ; i ∂ α X σ ∂ α X i − η µν F Bνσ ; i ǫ αβ ∂ α X σ ∂ β X i + O ( F A , F B , F A F B ) = 0 (A.6)These equations can be solved iteratively in the field strengths F Aµν and F Bµν . We need the solutionsat the zeroth order in the field strengths. However, we note the structure of the solution upto the28inear order which can be obtained to be X i ( τ , σ ) = X iR ( σ − ) + X iL ( σ + ) − g ij F Bµν ; j X µR ( σ − ) X νL ( σ + ) + O ( F A , F B , F A F B ) X µ ( τ , σ ) = X µR ( σ − ) + X µL ( σ + ) + 12 η µσ F Aσλ ; i X λ ( τ , σ ) X i ( τ , σ )+ 12 η µσ F Bσν ; i (cid:16) X νL ( σ + ) X iR ( σ − ) − X νR ( σ − ) X iL ( σ + ) (cid:17) + O ( F A , F B , F A F B ) (A.7)where σ ± = τ ± σ . The combinations X iR + X iL and X µR + X µL satisfy the Laplace equation. In thiswork, we shall not use this more general solution since only the leading order terms are required inthe expression of the Hamiltonian which gives the interaction of the string with the external gaugefields. By imposing the boundary conditions, the leading order terms can be expressed as X iR ( τ − σ ) = 12 x i + 2 πα ′ ℓ g ij p jR ( τ − σ ) + i r α ′ ∞ X n = −∞ n =0 n α in e − iπn ( τ − σ ) /ℓ (A.8) X iL ( τ + σ ) = 12 x i + 2 πα ′ ℓ g ij p jL ( τ + σ ) + i r α ′ ∞ X n = −∞ n =0 n ˜ α in e − iπn ( τ + σ ) /ℓ (A.9)The left and right moving momenta in the above expression are given by [22, 34] p iL = 12 √ α ′ (cid:20) n i + ( B ik + g ik ) m k (cid:21) , p iR = 12 √ α ′ (cid:20) n i + ( B ik − g ik ) m k (cid:21) (A.10)The X µR ( τ − σ ) and X µL ( τ + σ ) have the same structure as in (A.8) and (A.9) but with p iL and p iR replaced by p µ / where p µ denotes the momenta of the state.For computing the expectation value of the interacting hamiltonian, following expressions willbe useful (denoting X µ ≡ X µR + X µL ) ∂ + X iL = πℓ √ α ′ X n ˜ α in e − πin ( τ + σ ) /ℓ , ∂ + X µ = πℓ √ α ′ X n ˜ α µn e − πin ( τ + σ ) /ℓ ∂ − X iR = πℓ √ α ′ X n α in e − πin ( τ − σ ) /ℓ , ∂ − X µ = πℓ √ α ′ X n α µn e − πin ( τ − σ ) /ℓ (A.11)where, we defined ˜ α i = √ α ′ p iL , α i = √ α ′ p iR , ˜ α µ = α µ = √ α ′ p µ (A.12)The Virasoro’s generators in the compactified theory and to leading order in the gauge fields ex-pansion are: L = α ′ p R + α ′ p + N − L = α ′ p L + α ′ p + ˜ N − N = P ∞ n =1 (cid:2) g µν α µ − n α νn + g ij α i − n α jn (cid:3) and with a similarexpression for the left modes of the closed string.The mass-shell condition turns out to be: α ′ M = α ′ g ij p Ri p Rj + N − α ′ g ij p Li p Lj + ˜ N − (A.13)which implies the following level matching condition: N − ˜ N = n i m i (A.14)We counclude this section by observing that the above details do not change the evaluation of theleft and right-moving string amplitudes. The only difference is that when considering non-trivialwinding one should distinguish left and right momenta and perform the reduction via the followingreplacements: p MR → ( 12 p µ , p iR ) , p ML → ( 12 p µ , p iL ) . (A.15)Focusing on the first Regge trajectory one can then use the open string generating function definedin (C.7), combine two such generating functions, one for left and another for right moving part ofthe closed string correlator, and impose level matching (A.14) when expanding the oscillators andobtaining the closed string amplitudes. A.2 Type II sigma model
Superstring theory in a non trivial background is described by the (1 , -supersymmetric non linearsigma model that realizes the embedding of the string world-sheet in a space-time with a nontrivial metric G MN and an anti-symmetric tensor field B MN . The super conformal covariant actiondescribing the supersymmetric sigma model in the superspace notation is given by (see for exampleRef. [24]) S T ypeII = 14 πα ′ Z d σdθ + dθ − h G MN (Φ) D + Φ M D − Φ N + B MN (Φ) D + Φ M D − Φ N i (A.16)The world-sheet bosons X M and the Majorana fermions ψ M are collected in the superfield Φ M ( σ, θ ) = X M ( σ ) + ¯ θψ M ( σ ) + 12 ¯ θθF M ( σ ) (A.17)where F M ( σ ) are auxiliary fields with no dynamics and they will be eliminated by using theirequations of motion. The Grassmann coordinates θ are two-component Majorana spinors and ¯ θ = θ t ρ . The components of the spinors are labelled with the indices A = − , + . Hence, ψ M = (cid:18) ψ M − ψ M + (cid:19) ; θ = (cid:18) θ − θ + (cid:19) (A.18)Our convention for the gamma matrices are ρ = (cid:18) − ii (cid:19) ; ρ = (cid:18) ii (cid:19) (A.19)30sing these, we find Φ M ( σ, θ ) = X M ( σ ) − iθ − ψ M + ( σ ) + iθ + ψ M − + iθ + θ − F M ( σ ) (A.20)The covariant derivative is defined by ( A = − , + ): D A = ∂∂ ¯ θ A − i ( ρ α θ ) A ∂ α = ⇒ ( D − = − i ∂∂θ + − θ + ∂ − D + = i ∂∂θ − + 2 θ − ∂ + (A.21)Here, we have used the convention ∂ ± = 12 ( ∂ ± ∂ ) ; ∂ = ∂ + + ∂ − ; ∂ = ∂ + − ∂ − (A.22)We then have D + Φ M = 2 θ − ∂ + X M + ψ M + + θ + F M + 2 iθ − θ + ∂ + ψ M − D − Φ M = − θ + ∂ − X M + ψ M − + θ − F M + 2 iθ + θ − ∂ − ψ M + (A.23)and hence (noting that ψ ± are Grassmann valued fields) D + Φ M D − Φ N = θ − θ + (cid:16) − ∂ + X M ∂ − X N − iψ M + ∂ − ψ N + − F M F N + 2 i∂ + ψ M − ψ N − (cid:17) + θ − (cid:16) ∂ + X M ψ N − − ψ M + F N (cid:17) + θ + (cid:16) ∂ − X N ψ M + + ψ N − F M (cid:17) + ψ M + ψ N − We can now simplify the two terms in the action. After some manipulation, they are given by G MN (Φ) D + Φ M D − Φ N = θ − θ + (cid:20) − G MN ∂ + X M ∂ − X N − iG MN (cid:16) ψ M + ∇ − ψ N + + ψ N − ∇ + ψ M − (cid:17) − G MN F M F N − iG QN Γ QP M ψ P − ψ M + F N + ∂ P ∂ Q G MN ψ P − ψ Q + ψ M + ψ N − (cid:21) and B MN (Φ) D + Φ M D − Φ N = θ − θ + (cid:20) B MN (cid:16) − ∂ + X M ∂ − X N − iψ M + ∂ − ψ N + − iψ N − ∂ + ψ M − (cid:17) +2 i∂ P B MN ψ P − ∂ + X M ψ N − + 2 i∂ P B MN ψ P + ∂ − X N ψ M + − iH MNP ψ M + ψ N − F P + ∂ P ∂ Q B MN ψ P − ψ Q + ψ M + ψ N − (cid:21) (A.24)where, we defined ∇ − ψ M + = ∂ − ψ M + + Γ MP Q ∂ − X P ψ Q + , ∇ + ψ M − = ∂ + ψ M − + Γ MP Q ∂ + X P ψ Q − (A.25)The equation of motion of the auxiliary field F M gives F M = i Γ MP Q ψ P + ψ Q − + i G MN H NSR ψ R + ψ S − (A.26)31sing the above solution, the terms containing F M , in the action, can be simplified as − G MN F M F N − iG MN Γ MRS ψ R + ψ N − F S − iH MNP ψ M + ψ N − F P = h G MN Γ MP Q Γ NRS + H MP Q Γ MRS + 14 H TP Q H T RS i ψ Q + ψ S + ψ P − ψ R − (A.27)Now, we have ∂ P ∂ Q B MN ψ P − ψ Q + ψ M + ψ N − + H MP Q Γ MLS ψ Q + ψ S + ψ P − ψ L − = 12 h ∂ P (cid:0) ∂ Q B SL − ∂ S B QL (cid:1) − Γ MP S H QML − Γ MP Q H MSL i ψ Q + ψ S + ψ P − ψ L − = 12 ∇ P H QSL ψ Q + ψ S + ψ P − ψ L − (A.28)and ∂ P ∂ Q G MN ψ P − ψ Q + ψ M + ψ N − + G MN Γ MP Q Γ NRS ψ Q + ψ S + ψ P − ψ R − = 12 " (cid:16) ∂ L ∂ S G MP − ∂ P ∂ S G ML − ∂ L ∂ M G SP + ∂ P ∂ M G SL (cid:17) + G QN (cid:16) Γ NLS Γ QP M − Γ NLM Γ QP S (cid:17) ψ S + ψ M + ψ L − ψ P − = − R SMLP ψ S + ψ M + ψ L − ψ P − (A.29)We also have − iB MN ψ M + ∂ − ψ N + + 2 i∂ P B MN ψ P + ψ N + ∂ − X N = iH NP M ψ P + ψ M + ∂ − X N − i∂ − ( B MN ψ M + ψ N + ) (A.30)and, − iB MN ψ N − ∂ + ψ M − + 2 i∂ P B MN ψ P − ψ N − ∂ + X M = − iH NP M ψ P − ψ M − ∂ + X N + i∂ + ( B MN ψ M − ψ N − ) Using the above results and performing the grassmann integrals, the action can be written as(ignoring the boundary terms) S = 14 πα ′ Z d σ (cid:20) G MN ∂ + X M ∂ − X N + 4 B MN ∂ + X M ∂ − X N + 2 iG MN ψ M + ˜ ∇ − ψ N + + 2 iG MN ψ N − ˜ ∇ + ψ M − + 12 ˜ R MNP Q ψ M + ψ N + ψ P − ψ Q − (cid:21) (A.31)where, we defined ˜ ∇ − ψ M + = ∂ − ψ M + + (cid:16) Γ MP Q − H MP Q (cid:17) ψ P + ∂ − X Q (A.32) ˜ ∇ + ψ M − = ∂ + ψ M − + (cid:16) Γ MP Q + 12 H MP Q (cid:17) ψ P − ∂ + X Q (A.33)32nd, ˜ R MNP Q = R MNP Q + 12 ∇ P H MNQ − ∇ Q H MNP + 14 H MRP H RQN − H MRQ H RP N (A.34)Upto the quadratic order, the action for the world-sheet field X M is exactly the same as in thebosonic theory. Hence, upto the orders of our interest, the solutions of the equations of motion forthese fields are given by the same expressions as in the closed bosonic case. For the ψ M ± fields, theequations of motion, instead, are given by ∂ ± ψ N ∓ + ˜Γ N ± P Q ψ P ∓ ∂ ± X Q − i R NLSP ψ L ∓ ψ S ± ψ P ± = 0 (A.35)For our purposes, we need to only solve these at the lowest order, namely ∂ ∓ ψ M ± = 0 . Moreover,we only need the solution for the non compact directions which are given by ψ µ + = r πα ′ ℓ X r ∈ Z + a ¯ ψ µr e − iπr ( τ + σ ) /ℓ , ψ µ − = r πα ′ ℓ X r ∈ Z + a ψ µr e − iπr ( τ − σ ) /ℓ (A.36)where, a = 0 for R sector and a = for the NS sector.Below, we note some results which are useful in computing the Hamiltonian of the compactifiedtheory. In this case also, upto linear order in the fields, the Christoffel symbols which contribute aresame as in the bosonic case. Also, only H µν i = F Bµν, i contributes. Hence, we have ˜Γ µ ± νi = 12 η µσ F Aνσ, i ± η µσ F Bσν, i , ˜Γ i + νρ = ± g ij F Bνρ, j
Next, the S M defined in (3.34) are computed to be (using the notation ψ MN ± = ψ M ± ψ N ± ) S µ = − i F Aµρ, i (cid:16) ψ ρi + + ψ ρi − (cid:17) − i F Bρµ, i (cid:16) ψ ρi + − ψ ρi − (cid:17) (A.37)and, S i = − i F Aµρ, i (cid:16) ψ ρµ + + ψ ρµ − (cid:17) + i F Bρµ, i (cid:16) ψ ρµ + − ψ ρµ − (cid:17) (A.38)The T M defined in (3.37) have the similar expressions except that we need to change the sign infront of ψ MN + terms. A.3 Heterotic sigma model
The Heterotic theory has world-sheet supersymmetry in the right moving sector. Consequently, theSigma model for the Heterotic string theory is described by the action [27, 35, 36] S = S [Φ] + S [Φ] + S [Φ , Λ − ] (A.39)where, S [Φ] = 24 πα ′ Z d σdθ − G MN (Φ) D + Φ M ∂ − Φ N (A.40)33 [Φ] = 24 πα ′ Z d σdθ − B MN (Φ) D + Φ M ∂ − Φ N (A.41) S [Φ , λ − ] = − i πα ′ Z d σdθ − g AB (Φ)Λ A − ( D + + A + (Φ)) BC Λ C − (A.42)and, Φ M ( σ, θ − ) = X M ( σ ) − iθ − ψ M + , D + = i ∂∂θ − + 2 θ − ∂ + (A.43) Λ A − ( σ, θ − ) = λ A − + θ − f A ( σ ) , A B + C = ( A M (Φ)) BC D + Φ M (A.44) Λ A − is anti-commuting. Moreover, only the antisymmetric part of the gauge field A M contributes.Hence, we can take ( A M ) BC = − ( A M ) CB . We shall now do the component expansion and expressthe action in terms of the physical component fields. For this, we note that D + Φ M = ψ M + + 2 θ − ∂ + X M , ∂ − Φ N = ∂ − X N − iθ − ∂ − ψ N + (A.45) G MN (Φ) = G MN ( X ) − iθ − ψ P + ∂ P G MN , B MN (Φ) = B MN ( X ) − iθ − ψ P + ∂ P B MN g AB (Φ) = g AB ( X ) − iθ − ψ M + ∂ M g AB , A M (Φ) = A M ( X ) − iθ − ψ P + ∂ P A M ( A + ) BC = ( A M ) BC ψ M + + 2 θ − ( A M ) BC ∂ + X M − iθ − ψ P + ψ M + ( ∂ P A M ) BC g AB Λ A − = g AB λ A − + θ − g AB f A − iθ − ψ M + λ A − ∂ M g AB D + Λ B − = if B + 2 θ − ∂ + λ B − (A.46)The three terms in the action can be expressed as πα ′ G MN (Φ) D + Φ M ∂ − Φ N = 24 πα ′ θ − h G MN ∂ + X M ∂ − X N + iG MN ψ M + ∂ − ψ N + − i∂ P G MN ψ P + ψ M + ∂ − X N i (A.47) πα ′ B MN (Φ) D + Φ M ∂ − Φ N = 24 πα ′ θ − h B MN ∂ + X M ∂ − X N + iB MN ψ M + ∂ − ψ N + − i∂ P B MN ψ P + ψ M + ∂ − X N i (A.48) − i πα ′ g AB Λ A − (cid:2) D + Λ B − + ( A + ) BC Λ C − (cid:3) = − i πα ′ h − g AB λ A − ∂ + λ B − + g AB λ A − ψ M + ( A M ) BC f C − g AB λ A − λ C − ( A M ) BC ∂ + X M + ig AB λ A − ψ P + ψ M + λ C − ( ∂ P A M ) BC + ig AB f A f B + g AB ( A M ) BC ψ M + λ C − f A + ψ M + λ A − ∂ M g AB f B − iψ M + λ A − ψ N + λ C − ( A N ) BC ∂ M g AB i (A.49)34ow, we have ig AB λ A − ∂ + λ B − + 2 ig AB λ A − λ C − ( A M ) BC ∂ + X M = 2 ig AB λ A − ˆ ∇ + λ B − where, ˆ ∇ + λ B − = ∂ + λ B − + ( ˆ A M ) BC λ C − ∂ + X M , ( ˆ A M ) BC = ( A M ) BC + 12 g BD ∂ M g DC (A.50)The terms involving f A can be simplified as − ig AB λ A − ψ M + ( A M ) BC f C + g AB f A f B − ig AB ( A M ) BC ψ M + λ C − f A − iψ M + λ A − ∂ M g AB f B = − ig AB f A h if B + 2( ˆ A M ) BC ψ M + λ C − i (A.51)The equation of motion of f A gives f A = i ( ˆ A M ) AC ψ M + λ C − (A.52)Substituting this solution in (A.51) gives − ig AB f A h if B + 2( ˆ A M ) BC ψ M + λ C − i = − g AB ( ˆ A M ) AC ( ˆ A N ) BD ψ M + ψ N + λ C − λ D − (A.53)Using the above results and equation (A.30), we can express the action as S = 14 πα ′ Z d σ h G MN ∂ + X M ∂ − X N + 4 B MN ∂ + X M ∂ − X N + 2 iG MN ψ M + ˜ ∇ − ψ N + +2 ig AB λ A − ˆ ∇ + λ B − + 12 F MN ; CD ψ M + ψ N + λ C − λ D − i (A.54)where ˜ ∇ − ψ N + is defined in (A.32) and F MN ; CD = ∂ M ( ˆ A N ) CD − ∂ N ( ˆ A M ) CD + ( ˆ A M ) CB ( ˆ A N ) BD − ( ˆ A N ) CB ( ˆ A M ) BD (A.55)For the compactification on T D , we again need the mode expansion of the world sheet fields.At the quadratic order in the world-sheet fields, the equations of motion for the X M are same asin the bosonic and Type II theories. Hence, upto the orders of our interest, we can use the sameexpressions for their mode expansion as obtained in these non linear sigma models. The equationsof motion of the ψ M + , instead, are modified with respect to the corresponding equation in the (1 , supersymmetric sigma model by terms containing the background gauge field. However in thefollowing, we shall be only interested in the solution of these equations to the zeroth order in thegauge field A M . Therefore, for ψ M + also, we can use the same mode expansion as in the type IItheory. Finally, the equation of motion for λ B − fields are ∂ + λ B − + ( ˆ A M ) BC λ C − ∂ + X M + 12 g BD ∂ M ( g CD ) λ C − ∂ + X M − i g BA F MN ; AC ψ M + ψ N + λ C − = 0 (A.56)35his shows that at the lowest order, λ A − satisfies the same equation as ψ M − in type II case. Hence,the solution for these are also given by the same expressions as for ψ M − .We again use the same compactification ansatz for G MN and B MN as in the bosonic and typeII theory. Hence, the expression for the components of ˜Γ M − NP are same as in the case of type IItheory. For S M defined in (3.50), we have (using the notation ψ MN + = ψ M + ψ N + and λ AB − = λ A − λ B − ) S µ = − i F µρ ; i ψ ρi + − i F Bρµ ; i ψ ρi + − i g AB ( ˆ A µ ) BC λ A − λ C − (A.57)and, S i = − i F µρ ; i ψ ρµ + + i F Bρµ ; i ψ ρµ + − i g AB ( ˆ A i ) BC λ A − λ C − (A.58)Again, the T M defined in (3.53) have the similar expressions except that we need to change thesign in front of ψ MN + terms. B Generalities on higher spin states
In the following, it will be convenient to use the following spinning polarisations: φ µ ...µ s ( p ) = 1 s ! u µ . . . u µ s , (B.1)in terms of the transverse polarization vector u µ depending on the momentum p . Upon using pointsplitting the above replacement allows to transform any spinning correlator into a polynomial in theLorentz invariant contractions of the vector polarisation u i associated with each external leg.Anticipating the results, we shall see that the general pattern of the super-string scatteringamplitude generating function can be given as A = B exp( C ) , (B.2)where exp( C ) is the same generating function entering the bosonic string case obtained in [31],while B is overall polynomial in the polarisations whose form appears to be completely specifiedsolely by the fermionic correlators entering the computation in the superstring case.It is useful to write down in terms of the polarisations (B.1) various cubic couplings which weshall extract from string theory. Focusing on the EM coupling of HS fields which reads: − ieA ν φ ∗ µ ...µ s ←→ ∂ ν φ µ ...µ s → es ! u · p ( u · u ) s , (B.3)where we remind the reader that u · p = 0 . Similar expression can be obtained for the non-minimalPauli coupling: ig es F µν φ ∗ µµ ...µ s φ νµ ...µ s → − e g s ( s − ( p µ u ν − p ν u µ ) u ,µ u ,ν ( u · u ) s − (B.4) = e g s s − ( u · p u · u + u · p u · u ) ( u · u ) s − , (B.5)36here we introduced the gyromagnetic factor g . Similar polynomial expressions can be obtained forany coupling by simply using (B.1).Here, we also introduce the representation of the spin-operator on totally symmetric states: ( S MN ) M ( s ) N ( s ) = 2 isη ( M [ M δ N ]( N . . . δ M s ) N s ) (B.6)where M ( s ) (and N ( s ) ) denotes the completely symmetrized set of indices { M . . . M s } and wedefine u s · S MN · u s ≡ u M . . . u M s ( S MN ) ( M ...M s )( N ...N s ) u N . . . u N s = 2 isu [ M u N ]3 ( u · u ) s − (B.7)So far, we have focused on higher spin states in open string theories. In closed string theories, thepolarization of the physical states can be obtained by the factorized product of holomorphic (right)and antiholomorphic (left) sectors, in the following denoted with u and ¯ u respectively. Two spinoperators, S µνR ; L , can be introduced in such theories and their action on the polarization of thehigher spin states can be equivalently defined as: S µνR = i (cid:0) u µ ∂∂u ν − u ν ∂∂u µ (cid:1) ; S µνR = i (cid:0) ¯ u µ ∂∂ ¯ u ν − ¯ u ν ∂∂ ¯ u µ (cid:1) (B.8)It is easily seen that : S µνR u µ . . . u µ l = 2 i l X i =1 δ [ νµ i η µ ] a i δ a µ . . . δ a i − µ i − δ a i +1 µ i +1 . . . δ a l µ l u a . . . u a l = 2 iη a [ µ l X i =1 δ ν ] µ i l − (cid:0) δ a ( µ . . . δ a i − µ i − δ a i +1 µ i +1 . . . δ a l µ l ) (cid:1) u a . . . u a l = 2 i lη a [ µ δ ν ]( µ . . . δ a l µ l ) u a . . . u a l (B.9)This coincides with Eq. (B.6).The completely symmetric closed string state is defined by the linear combination φ µ ...µ l ν ...ν k = u ( µ . . . u µ l ¯ u ν . . . ¯ u ν l ) (B.10)The action of the spin operators on such states gives: S µνR φ µ ...µ l ν ...ν k = 2 i l η a [ µ δ ν ]( µ . . . δ a l µ l δ a l +1 ν . . . δ a l + k ν k ) u a . . . u a l ¯ u a l +1 . . . ¯ u a l + k S µνL φ µ ...µ l ν ...ν k = 2 i k η a l +1 [ µ δ ν ]( µ . . . δ a l µ l δ a l +1 ν . . . δ a l + k ν k ) u a . . . u a l ¯ u a l +1 . . . ¯ u a l + k (B.11)and ( S µνL + S µνR ) φ µ ...µ l ν ...ν k = 2 i ( l + k ) η ( a [ µ δ ν ]( µ . . . δ a l µ l δ a l +1 ν . . . δ a l + k ) ν k ) u a . . . u a l ¯ u a l +1 . . . ¯ u a l + k (B.12) The notation is u ( µ . . . u µ l ) = l ! P perm. { ...l } u µ . . . u µ l = u µ . . . u µ l ( w, ¯ w ) , the totally symmetric state can berepresented in the compact form: φ k + l = N l + k ( u · w ) k ( u · ¯ w ) k (B.13)where the left and right polarizations are now identified to get a totally symmetric tensor. Here N l + k is an overall normalization factor that has to be fixed by requiring that the amplitudes givecanonically normalized kinetic terms.In this representation of spinning particles, the actions of the S R and S L spin operators definedin Eq.s (B.12) are obtained by acting on the states with the operators: S µνR = i (cid:16) w µ ∂∂w ν − w ν ∂∂w µ (cid:17) ; S µνL = i (cid:16) ¯ w µ ∂∂ ¯ w ν − ¯ w ν ∂∂ ¯ w µ (cid:17) (B.14)introduced in Section 2. C Summary of open string amplitudes for the bosonic statesin the first Regge trajectory
In this section we list the open bosonic three point amplitude in bosonic and super-string in thenon compact and compact background. Closed string amplitudes can be obtained as appropriatesquares of the open-string amplitudes.
C.1 Generating Functions for Vertex operators
Along the lines of [31] we can use the replacement (B.1) to construct vertex operator generatingfunctions bosonic string state and NS or R superstring states. Performming the same replacementone can further resum in a single function all vertex operators with different spins. The bosonicvertex operators after resummation in terms of the vector polarisation (B.1) then reads V ( k R , u ) = exp (cid:16) i∂X · u + i √ α ′ k R · X (cid:17) , (C.1) k R (and k L in the closed string) is the momentum of the string state and the label R is irrelevant inopen-string but in the closed string, where the vertices are the product of the right (or holomorphic)and left (antiholomorphic) sectors, denotes the right-momentum of the particle. In non compactspaces, k R ≡ p and k R = k L ≡ p/ in open and closed string , respectively. In the compact toroidalbackground, instead, the momentum long the compact directions is quantized being in the closedstring equal to k R, L ≡ ( p/ , p R, L ) with p R, L defined in Eq. (A.10). In Eq.(C.1), we have extracted from the string field X the overall normalization factor √ α ′ to make such aquantity dimensionless. − can then be given as V ( − ( k R , u ) = : ψ · u e − φ exp (cid:16) i∂X · u + i √ α ′ k R · X (cid:17) : , (C.2)where the dependence on the world-sheet coordinate is understood.Similarly, one can construct a generating function of NS vertex operators in the non-canonical picture with ghost charge as V (0) ( k R , ξ ) =: nh ( ∂ψ · u ) + √ α ′ k R · ψ i ( ψ · u ) + u · ∂ u o exp (cid:16) i∂X · u + i √ α ′ k R · X (cid:17) : , (C.3)In the following we will use the vertex operator generating functions so far obtained in order tocompute superstring scattering amplitudes. Notice that the transversality condition for the polar-ization tensors can be translated into a transversality condition for the polarisation u that from nowon will be projected on its transverse components. Moreover all vertex generating functions areproportional to the same exponential factor exp (cid:16) i∂X · u + i √ α ′ k · X (cid:17) , (C.4)Finally it is useful to comment on closed string vertex operators. The key observation is thatthese can be obtained multiplying together holomorphic and anti-holomorphic open string verticesin the corresponding pictures and imposing level matching: V ( n,m ) ( k R , k L , u, ¯ u ) = L (cid:2) V ( n ) ( k R , u ) V ( m ) ( k L , ¯ u ) (cid:3) (C.5)An explicit implementation of the level matching operation is given by: L [ f ( u ) g (¯ u )] = I (cid:16) p λ ¯ λ (cid:17) f ( λu ) g (¯ λ ¯ u ) (cid:12)(cid:12)(cid:12) λ =¯ λ =0 . (C.6) I is the Bessel function.When imposing level matching the normalisation of the polarisation tensor is not anymore theone given in (B.1). It is straightforward to fix this normalisation but it is not necessary for thepurposes of these notes. C.2 Summary of open string amplitudes for the bosonic states in thefirst Regge trajectory
In this section we just list the open bosonic three point amplitude in bosonic and super-string in thenon compact and compact background. Closed string amplitudes can be obtained as appropriatesquares of the open-string amplitudes. 39 osonic Generating Function
The bosonic generating function of three point amplitudes isobtained by evaluating the string correlator with three vertices defined in Eq.(C.1). This containsan explicit dependence on the tree level string Green function. However, three point amplitudes,on-shell, don’t have any dependence on such quantities and we can write the following expressionfor the generating function: B ( p i , u i ) = e u · u + u · u + u · u − r α ′ u · p + u · p + u · p ) (C.7)with p ij = p i − p j and we have used the on-shell condition p i · u i = 0 . Superstring Generating Function
The superstring generating function of three point ampli-tudes is instead obtained by evaluating the correlator with two vertices defined in Eq. (C.2) andone vertex given in Eq. (C.3), one gets: S ( p i , u i ) = (cid:16) G − u · u u · u − u · u u · u − u · u u · u (cid:17) (C.8) × e u · u + u · u + u · u − r α ′ u · p + u · p + u · p ) where we have defined the YM combination G = r α ′ u · u u · p + u · u u · p + u · u u · p ] (C.9)Closed string amplitude can be obtained from products of open string amplitudes enforcing levelmatching. We can obtain both closed bosonic, super and heterotic 3pt amplitudes just implementinglevel matching when expanding the generating functions.In order to obtain the dimensional reduction on the D-dimensional torus we write the polar-ization tensor in the form u → ( u, v ) being u and v the non compact and compact components,respectively. The amplitude in the compact space is obtained from the corresponding one in thenon compact background by implementing dimensional reduction rules. These in open string are: u i · u j → u i · u j + v i · v j (C.10) u i · p j → u i · p j + v i · p j (C.11)plus cyclic, where p j = n j R being n j the KK level and R the compactification length. The corre-sponding rules in closed-string are defined in Eq. (C.17).In the following we shall extract the part of the coupling which involves one derivative andcompare it with (B.5) extracting the corresponding gyromagnetic factor g . This can be obtainedby taylor expanding the generating functions presented in this section. This is straightforward usingthe series expansion of exponential. 40 pen bosonic string The scattering amplitude with one massless state interacting with twohigher-spin states of the leading Regge-trajectory is obtained by expanding the exponential in Eq.(C.7) and keeping the terms linear in photon polarization and of order u s . The expression alleading order in the string slope turns out to be: s ! ( u · u ) s − r α ′ u · p u · u + s ( u · p u · u + u · p u · u )]1 s ! ( u · u ) s − r α ′ h u · p u · u − s ( u M p N − u N p M ) u [ M u N ]2 i (C.12)Here, u is the photon polarization while u , are the polarizations of the massive particles. Eq.(B.7)allows to write Eq.(C.12) as follows: s ! r α ′ (cid:20) u · p ( u · u ) s + i u M p N − u N p M ) u s · S MN · u s (cid:21) (C.13)which is equivalent to g = 2 . From an effective field theory approach to string amplitudes, thefirst term of this equation comes from the kinetic term of the higher spin states minimally coupledwith an abelian field. The normalization of the string vertices has to be fine-tuned to get canon-ically normalized kinetic terms. We don’t write down explicitly such normalization but it can bestraightforward read from Eq. (C.13). In terms of the physical polarization defined in Eq. (B.1)the three-point amplitude takes the form: A ∼ h u p φ · φ + 12 F MN φ a ( s ) ( S MN ) a ( s ) b ( s ) φ b ( s )2 i (C.14)with F MN defined in Eq.(4.7). Open superstring string
Similarly, the superstring amplitude involving one-massless state andtwo higher spins of the leading Regge trajectory is: s − u · u ) s − r α ′ u · p u · u + s ( u · p u · u + u · p u · u )] (C.15)which is again and as expected to g = 2 . This amplitude when written in terms of the polarizationof the higher spin states given in Eq. (B.1) coincides, at the leading order in the string slope, withEq. (C.14). Closed strings after reduction on a torus T D The closed amplitudes before the compactifi-cation are the same in bosonic, superstring and heterotic string, therefore, being equal to: A cl. ∼ (cid:20) u · k R ;23 ( u · u ) s R + i u M k R ;1 N − u N k R ;1 M ) u s R · S MNR · u s R (cid:21) × (cid:20) ¯ u · k L ;23 (¯ u · ¯ u ) s L + i u M k L ;1 N − ¯ u N k L ;1 M ) ¯ u s L · S MNL · ¯ u s L (cid:21) (C.16)41he compactification is easily performed by implementing the reduction rules given in Eq.s (C.11).These correspond to the replacements u · k r, ( u · u ) s R → (cid:16) u · p + v · p R ;23 (cid:17) ( u · u + v · v ) s R ( u M k R ;1 N − u N k R ;1 M ) u s R · S MNR · u s R → is R ( u µ p ν − u ν p µ ) u [ µ u ν ]2 ( u · u + v · v ) s R − +2 is R v a p ν ( v a u ν − u ν v a )( u · u + v · v ) s R − (C.17)Similar relations hold for the antiholomorphic sector.In the following, we consider in the amplitude only terms relevant for the determination of thegyromagnetic ratio of the massive fields with respect to the U (1) -gauge fields described by thepolarization tensors of the form ε (1) µa ≡ u µ ¯ v a and ε (1) aµ ≡ v a ¯ u µ . Therefore, we ignore, in thereduction, terms like the last line of Eq.(C.17) and those quadratic in the non-compact momentathat correspond to couplings with two derivatives. The reduced amplitude turns out to be: A cl. ∼ h (cid:16) p µ ε µa p aL ;23 + p aR ;23 ε aµ p µ (cid:17) ( u · u + v · v ) s R (¯ u · ¯ u + ¯ v · ¯ v ) s L + i s L ( ε aµ p ν − ε aν p µ ) p R ;23 ¯ u [ µ ¯ u ν ]2 ( u · u + v · v ) s R (¯ u · ¯ u + ¯ v · ¯ v ) s L − + i s R ( ε µa p ν − ε νa p µ ) p L ;23 u [ µ u ν ]2 ( u · u + v · v ) s R − (¯ u · ¯ u + ¯ v · ¯ v ) s L . . . i (C.18)By introducing the U (1) -fields: ε νa = A µa + B µa ; ε aµ = A µa − B µa (C.19)with their field strength F Aµν ; a = i ( p µ A ν ; a − p ν A µ ; a ) ; F Bµν ; a = i ( p µ B ν ; a − p ν B µ ; a ) (C.20)the amplitudes is rewritten in the form: A cl. ∼ " ( Q a A a · p + Q a B a · p ) ( u · u + v · v ) s R (¯ u · ¯ u + ¯ v · ¯ v ) s L + 12 F Aµν ; a (cid:16) p aL ;2 i s R u [ µ u ν ]2 (¯ u · ¯ u + ¯ v · ¯ v ) + p aR ;2 i s L ¯ u [ µ ¯ u ν ]2 ( u · u + v · v ) (cid:17) × ( u · u + v · v ) s R − (¯ u · ¯ u + ¯ v · ¯ v ) s L − + 12 F Bµν ; a (cid:16) p aL ;2 i s R u [ µ u ν ]2 (¯ u · ¯ u + ¯ v · ¯ v ) − p aR ;2 i s L ¯ u [ µ ¯ u ν ]2 ( u · u + v · v ) (cid:17) × ( u · u + v · v ) s R − (¯ u · ¯ u + ¯ v · ¯ v ) s L − + . . . (C.21)being: Q = p L ;2 + p R ;2 = − p L ;3 − p R ;3 ; Q = p L ;2 − p R ;2 = − p L ;3 + p R ;3 (C.22)42he charges of the two gauge fields.The binomial expansion allows to separate in the amplitude fields with different spin with respectto the reduced d − D -Lorenz group , it gives: ( u · u + v · v ) s R (¯ u · ¯ u + ¯ v · ¯ v ) s L = ( v · v ) s R (¯ v · ¯ v ) s L + h s L X k =0 s R X l =1 + s L X k =1 s R X l =0 i(cid:16) s L k (cid:17)(cid:16) s R l (cid:17) (¯ u · ¯ u ) k ( u · u ) l (¯ v · ¯ v ) s L − k ( v · v ) s R − l (C.23)We now define. φ φ = 1Γ( s R + 1)Γ( s L + 1) ( v · v ) s R (¯ v · ¯ v ) s L (C.24)and φ l + k · φ l + k = (¯ u · ¯ u ) k ( u · u ) l k ! l ! (¯ v · ¯ v ) s l − k ( v · v ) s R − l ( s L − k )! ( s R − l )! (C.25)Whit this definition of scalar product we get canonically normalized kinetic terms, being: ( u · u + v · v ) s R (¯ u · ¯ u + ¯ v · ¯ v ) s L = Γ( s R + 1)Γ( s L + 1) h φ φ + h s L X k =0 s R X l =1 + s L X k =1 s R X l =0 i φ k + l · φ k + l (C.26)In the same way: i s R u [ µ u ν ]2 ( u · u + v · v ) s R − (¯ u · ¯ u + ¯ v · ¯ v ) s L = 2 i s R − X l =0 s L X k =0 s R ( s R − s R − − l )! l ! s L !( s L − k )! k ! × u [ µ u ν ]2 (¯ u · ¯ u ) k ( u · u ) l (¯ v · ¯ v ) s L − k ( v · v ) s R − − l = s R X l =1 s L X k =0 (cid:16) s L k (cid:17)(cid:16) s R l (cid:17) × i l u [ µ u ν ]2 ( u · u ) l − (¯ u · ¯ u ) k (¯ v · ¯ v ) s L − k ( v · v ) s R − l = Γ( s R + 1)Γ( s L + 1) s R X l =1 s L X k =0 φ l + k · S µνR · φ l + k (C.27)Similar relation hold for S µνL . The amplitude can be written as: A cl. ∼ s R X l =0 s L X k =0 h ( Q a A a · p + Q a B a · p ) φ l + k · φ l + k + 12 F Aµν ; a (cid:16) p aL ;2 φ l + k S µνR · φ l + k + p aR ;2 φ l + k S µνL · φ l + k (cid:17) + 12 F Bµν ; a (cid:16) p aL ;2 φ l + k S µνR · φ l + k − p aR ;2 φ l + k S µνL · φ l + k (cid:17) . . . i (C.28)where we have used the identity: φ k S µνR · φ k = φ l +03 S µνL · φ l +02 = 0 (C.29) d = 10 or in superstring and bosonic string, respectively. D can be taken arbitrary. A ∼ s R X l =0 s L X k =0 n ( Q a A a · p + Q a B a · p ) φ l + k · φ l + k + 14 F Aµν ; a φ l + k · h Q a ( S µνR + S µνL ) + Q a ( S µνR − S µνL ) i · φ l + k + 14 F Bµν ; a φ l + k · h Q a ( S µνR − S µνL ) + Q a ( S µνR + S µνL ) i · φ l + k . . . o (C.30) D Young Tableaux and Polynomials
In this section, we shall give some details about the mixed-symmetry representations considered inthis work. For simplicity, we shall work with projectors and in particular with products of Kronecker- δ ’s which we shall label conveniently as δ b ( s ) ...b n ( s n ) a ( s ) ...a n ( s n ) . (D.1)Indices are appropriately projected onto a given mixed-symmetry representation. To each of theabove projector one can associate a polynomial built out of auxiliary variables u i and w i which playthe role of dummy variables associated to each set of totally symmetric indices. For instance thetotally symmetric projector reads T ℓ ( u | w ) = N ℓ ( u · w ) ℓ , (D.2)where N ℓ is a normalisation factor which can be fixed by requiring that the above projector squaresto itself under contraction of indices, namely N ℓ ℓ ! ( ∂ w · ∂ w ) ℓ ( u · w ) ℓ ( u · w ) ℓ = N ℓ ( u · u ) ℓ , (D.3)implying N ℓ = 1 ℓ ! . (D.4)It is convenient to normalise the contraction of indices for each set of totally symmetric indices as: Y i ℓ i ! ( ∂ w i · ∂ ¯ w i ) ℓ i , (D.5)and define the inner product f ( w i ) ◦ w g ( w i ) = X ℓ i Y i ℓ i ! ( ∂ w i · ∂ w i ) ℓ i f ( w i ) g ( ¯ w i ) (cid:12)(cid:12)(cid:12) w i = ¯ w i =0 . (D.6)44 L + ℓ R Figure 1: Young diagram for totally symmetric fieldsSimilar projectors can be constructed also for mixed-symmetry fields. For instance the hook projectorreads T ℓ, ( u , u | w , w ) = N ℓ, ( u · w ) ℓ − ( u · w u · w − u · w w · w ) , (D.7)where, the constant N ℓ, can be fixed to be: N ℓ, = 1 ℓ + 1 1( ℓ − , (D.8)by requiring that T ℓ, ( u , u | w , w ) ◦ w T ℓ, ( w , w | v , v ) = T ℓ, ( u , u | v , v ) . (D.9)The irreducibility condition is manifestly satisfied and takes the form u · ∂ u T ℓ, ( u , u | w , w ) = 0 = w · ∂ w T ℓ, ( u , u | w , w ) . (D.10)Sometime, with some abuse of notation, it can be convenient to represent the above projectorsusing a Dirac notation like: T ℓ ( u | w ) = | u i ℓ ℓ h w | , (D.11) T ℓ, ( u , u | w , w ) = | u , u i ℓ, ℓ, h w , w | . (D.12)In this work we need to project the tensor product of two totally symmetric representation intoirreducible components to extract the corresponding gyromagnetic factors. This is equivalent todecompose into irreducible components the polynomial ( u · w ) ℓ ( u · w ) ℓ and in particular to findthe contribution of a given representation to such inner-product. Similar decomposition problemscan be worked out more generally but they can be addressed similarly and for this reason will notbe considered here.To proceed, we need to define intertwiner operators projecting the tensor product of two totallysymmetric representations into the possible irreducible representation. Starting from the tensorproduct of two totally symmetric states, the simplest intertwiner projects onto the totally symmetriccomponent. This gives I ℓ L + ℓ R ℓ L ,ℓ R ( u, ¯ u | w ) = N ℓ L + ℓ R ℓ R ,ℓ L ( u · w ) ℓ L (¯ u · w ) ℓ R = ( | u i ℓ L ⊗ | ¯ u i ℓ R ) ℓ L + ℓ R h w | , (D.13)45hose normalization can be obtained by requiring I ℓ L + ℓ R ℓ L ,ℓ R ( u , u | w ) ◦ u I ℓ L + ℓ R ℓ L ,ℓ R ( u , u | v ) = T ℓ L + ℓ R ( v | w ) , (D.14)which gives N ℓ L + ℓ R ℓ R ,ℓ L = s ℓ ! ℓ !( ℓ + ℓ )! . (D.15)With the above intertwiner operators, it is straightforward to evaluate the string inner product h Φ | xS L + yS R | Φ i α , (D.16)expressing it in terms of the gyromagnetic factors introduced in §
2. To do this, it is sufficient toevaluate the action of S L,R as h Φ | xS ( u ) L + yS (¯ u ) R | Φ i α = I ℓ L + ℓ R ℓ L ,ℓ R ( w | u, ¯ u ) ◦ u h ( xS ( u ) L + yS (¯ u ) R ) I ℓ L + ℓ R ℓ L ,ℓ R ( u, ¯ u | v ) i = T ℓ + ℓ ( w | u ) ◦ u [ α S ( u ) T ℓ + ℓ ( u | v )]= h Φ | α S | Φ i u , (D.17)where the coefficient α can be easily extracted to be α = xℓ + yℓ ℓ + ℓ , (D.18)which proves eq. (5.9).To obtain the projection of the closed string states into mixed-symmetry components it is againsufficient to derive the corresponding intertwiner operators. In the case of the hook field, one hassimply I ℓ L + ℓ R − , ℓ L ,ℓ R ( u, ¯ u | w , w ) = ¯ N ℓ L + ℓ R − , ℓ L ,ℓ R ( u · w ) ℓ − (¯ u · w ) ℓ − ( u · w ¯ u · w − u · w ¯ u · w ) , (D.19)The normalisation can again be obtained by requiring I ℓ L + ℓ R − , ℓ L ,ℓ R ( u, ¯ u | w , w ) ◦ u I ℓ L + ℓ R − , ℓ L ,ℓ R ( u, ¯ u | v , v ) = T ℓ L + ℓ R − , ( w , w | v , v ) , (D.20)which gives N ℓ L + ℓ R − , ℓ L ,ℓ R = 1 ℓ L + ℓ R s ℓ L − ℓ R − ℓ L + ℓ R − . (D.21)It is now straightforward to evaluate the string-inner product on the intertwiner operator andrewrite it in terms of the canonical spin operators acting on the mixed symmetry representation.The result reads I ( ℓ L + ℓ R − , ℓ L ,ℓ R ( u, ¯ u | w , w ) ◦ u h ( xS ( u ) L + yS (¯ u ) R ) I ( ℓ L + ℓ R − , ℓ L ,ℓ R ( u, ¯ u | v , v ) i = T ℓ L + ℓ R − , ( u , u | w , w ) ◦ u h ( α S ( u )1 + α S ( u )2 ) T ℓ L + ℓ R − , ( u , u | v , v ) i , (D.22)46 + ℓ − kk Figure 2: Young diagram with two rowswhere in the second line we used u and u instead of u and ¯ u to indicate the mixed symmetrydummy variables. The above calculation then gives α = ( ℓ − x + ( ℓ − yℓ + ℓ − , (D.23) α = ( ℓ − y + ( ℓ − xℓ + ℓ − . (D.24)These results can be generalised with some effort to the most general case. For example, focusingon the first Regge trajectory of the closed string, we can consider the irreducible projection into anarbitrary two row Yang tableaux of the type { ℓ + ℓ − k, k } . In this generic case we have T ℓ,k = 1( ℓ − k )! k !( ℓ − k + 2) k ( u · w ) ℓ − k ( u · w u · w − u · w u · w ) k , (D.25)together with the intertwiner I ℓ + ℓ − k,kℓ L ,ℓ R ( u, ¯ u | w , w ) = 1 k !( ℓ + ℓ + 2 − k ) k s ℓ − k )!( ℓ − k )!( ℓ + ℓ − k )! × ( u · w ) ℓ − k (¯ u · w ) ℓ − k ( u · w ¯ u · w − u · w ¯ u · w ) k . (D.26)With the above tensor, it is tedious but straightforward to evaluate the inner product betweenclosed string states as I ℓ + ℓ − k,kℓ L ,ℓ R ( u, ¯ u | w , w ) ◦ u h ( xS ( u ) L + yS (¯ u ) R ) I ℓ + ℓ − k,kℓ L ,ℓ R ( u, ¯ u | v , v ) i = T ℓ + ℓ − k,k ( w , w | u , u ) ◦ u h ( α S ( u )1 + α S ( u )2 ) T ℓ + ℓ − k,k ( v , v | u , u ) i , (D.27)so that one can find the value of the coefficients α , α = x ( ℓ − k ) + y ( ℓ − k ) ℓ + ℓ − k (D.28) α = y ( ℓ − k ) + x ( ℓ − k ) ℓ + ℓ − k , (D.29)47hich is valid for any k , ℓ and ℓ for which a corresponding representation exists. This is the resultgiven in equation (5.13).The case of square tableaux needs to be addressed separately or through a limiting procedure.With the above results in hand, we can obtain the gyromagnetic ratios for arbitrary two rowrepresentations in the first Regge trajectory of the closed bosonic string. Note that for k = 0 , theabove results neatly reduce to the special cases discussed above. References [1] M. Fierz, W. Pauli, O n relativistic wave equations for particles of arbitrary spin in an electro-magnetic field, Proc.Roy.Soc.Lond. A 173 (1939) 211-232.[2] F. J. Belifante,
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