JHEP07(2009)073 F Nt July 20, 2009 June 28, 2009 and Baryons April 28, 2009 : : : Published Received Accepted Son Is Represented by an and Hence Degenerate

JHEP07(2009)073 f nt July 20, 2009 June 28, 2009 and baryons April 28, 2009 : : : Published Received Accepted son is represented by an and hence degenerate. To y an un-oriented fermionic 10.1088/1126-6708/2009/07/073 s is in full agreement with the tion of mesons and baryons as approximation” asymptotically doi: N Published by IOP Publishing for SISSA ) theory with a quark in the two-index antisym- N [email protected] , , due to an equivalence with super Yang-Mills, the tensions o N 0901.4508 1/N Expansion, QCD, Supersymmetric gauge theory, Conﬁneme We consider the degeneracy between the Regge slope of mesons [email protected] SISSA 2009 Department of Physics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K. E-mail: c

Abstract: Adi Armoni and Agostino Patella in QCD. We argue that within the “orientifold large- Degeneracy between the Reggebaryons slope from of supersymmetry mesons and massive mesons and baryonsthis become end, supersymmetric we partners generalize QCD by a SU( ArXiv ePrint: the bosonic and theoriented fermionic and strings un-oriented coincide. bosonic Our andspectra descrip fermionic of QCD-string open strings in the dual type 0’Keywords: string theory. metric representation. Weoriented show bosonic that QCD-string in and thisQCD-string. the framework baryon the At is me large- represented b JHEP07(2009)073 e 1 2 7 7 8 ]) 10 2 (1.1) = 1 Super N ) N at both mesons and ) gauge theory with a s not clear whether a e of the Regge slopes of N s. y and why the Regge slope U( e a review by Wilczek [ nteraction, the low-energy ent, we are confident that upled, remains notoriously rk pair behaves like a string e: the baryon is composed planar equivalence”. ector to SU( n (commonly called “orientifold , 2 M ′ α ), or whether it holds for SU(3) only. + N 0 – 1 – α and hence an intuitive picture of the Regge phe- ] for a recent work). A fastly rotating relativistic 1 = J πσJ = 2 2 M is called the Regge slope. A Regge trajectory for mesons can b 2 − 89 GeV . This string is often called the QCD-string. . ′ 0 1 πα 2 ∼ ′ = α σ “Orientifold planar equivalence” is the statement that an S In this paper we propose a new explanation for the coincidenc One of the interesting aspects of the hadronic spectrum is th It is less obvious why baryons admit a linear Regge trajector massless Dirac fermion in thefield antisymmetric theory”) representatio becomes equivalent in a certain well-defined s easily explained by string theory (see [ where baryons and mesons. Our derivation is based on “orientifold (infra-red) regime, wheredifficult. quarks and Despite gluons aquarks are lack are confined of strongly by analytical co a proof strong force of to color form confinem mesons and baryon 1 Introduction Despite of the great success of QCD in describing the strong i C SYM in the Hamiltonian formalism 3 Conclusions A Review of gauge theories inB the Hamiltonian Static formalism heavy fundamental charges baryons fall into a linear Regge trajectory of the form Contents 1 Introduction 2 Meson and baryons in “orientifold field theories” string admits the relation is that the baryonof structure a is quark similargeneralization and to of a this the diquark picture meson applies connected structur to by SU( a single long string. It i of mesons and baryons is the same. A possible explanation (se nomenon is that the fluxwith tube that connects the quark anti-qua JHEP07(2009)073 3 ]. ], ). of 6 7 Q dex . ym- (2.1) ) ]. We 1 N > x 13 ( i an unbro- Q rk and two i l ) with N d~y ) y ( ~ A string which contains 1 mond model [ x aryon in the “orientifold z ture there is a smeared etric representation are rk pair connected by a ] and references therein. Z 9 persymmetric theory [ in the fundamental repre- i ture in the sense that there SU(2), as the fundamental degenerate via the relation Q sive quarks (denoted by to estimate non-perturbative × s ]. A related, however different, N e there is one quark at one end 6 amiltonian formalism, therefore m SU(3) to SU( ion of the quark condensate [ Pexp ) z ( ] theories”, see [ kl [ β ] in order to explain spectra degeneracies ¯ Ψ 12 αβ – ) We show that the two QCD-strings become ~γ 1 = 1 super Yang-Mills. It led to various analyt- 10 j k x 1 – 2 – ( i N Q d~y ) i j y ]. Provided that the equivalence holds it enables us ( 5 ~ A d~y ) z 0 y x ( Z ~ i A 1 . Therefore, in our picture, the meson and the baryon Regge 0 x x N Z for a review of the main concepts) Pexp i ) gauge theory with one massless quark in the two-index antis A the dynamics of the model becomes equivalent to the dynamics ) N 0 ) gauge theory with one massless Dirac fermion Ψ in the two-in x Pexp ( N C j ]. A necessary and sufficient condition for the equivalence is → ∞ ]. Moreover, for SU(3) it reduces to QCD with one massless qua ) 4 0 , 14 x N ]. 3 ( d~zQ 8 j ¯ Q Z = = α Throughout the paper we will use a version of the Corrigan-Ra The idea of the present paper is to describe the meson and the b Our picture of the baryon is similar to the quark-diquark pic There are other generalizations of baryons in “orientifold M = 0; see appendix. B = 1 SYM [ 1 0 slopes are degenerate due to supersymmetric relics in QCD [ is smooth, since even for SU(3) the flavor symmetry is U(1) ical non-perturbative results in QCD, including a calculat Yang-Mills [ Moreover, since forequivalent SU(3) to each the other, fundamentalquantities planar in and equivalence QCD provides the by a copying anti-symm them way from see review [ between mesons and baryons. will consider an SU( additional massive quarks. Note that the interpolation fro to copy results from the supersymmetric theory to the non-su superpartners at infinite ken charge conjugation symmetry [ “effective supersymmetry” was assumed in [ theory” by operators whose mass spectra become at large- with super Yang-Mills.QCD-string and Briefly, the the baryonan meson is anti-symmetric is a fermion quark a smeared quark quark on pair it. anti-qua connected by a is only one longquark QCD-string along the in string, the whereasof in baryon. the the string quark-diquark However, and pictur in two quarks our at pic the other end. 2 Meson and baryons inConsider “orientifold an field SU( theories” quarks are massive. metric representation (denoted by Ψ) and two additional mas In the limit N sentation. Let us consider theA following operators (in the H antisymmetric representation and one or more Dirac fermion JHEP07(2009)073 α (2.2) (2.3) (2.4) 3, the ¯ Ψ can for the / ]. 1 B − 13 3 and the Hamiltonian and we use them / ppendix tring-breaking en- is an intermediate ]), we will be able Q , the fermion z ] 15 . jk [ ]): B ¯ ic number of QCD. This Ψ l representation, and the s made of light quarks is is a fermionic operator ( uarks limit, therefore the string 15 . In analogy with mesons, nic string tensions. Light heavy quarks do not follow ijk B ope is related to the tension ǫ N ell. Microscopically it can be and - 2 1 nd Ramond in ref. [ ) (2.5) as a generalization of ordinary 0 c semiclassical quark-antiquark tring. This description implies a M = R B ( i q creates a baryonic state, built with b O i . and + B of the string tensions. In fact charges of the heavy quarks, the energy states . b Q . is the charge-conjugation matrix acting n, r i are effectively described by a relativistic | b M f for both R N C 1 B nr 1 πσ n, r y πσ | 2 2 M R = – 3 – b n = nr and where σ X ′ B ′ M x α C α ) (but the mass gets a shift [ = = limit the two string tensions are equal, proving this T b i is a bosonic operator and i 1.1 , we decompose the states obtained with the operators 0 Q N | and Ψ respectively baryonic numbers 1 Q M = n, r M , Q | C N ¯ H Ψ. We think about Q of the bosonic string (see for instance [ b σ : R and a fermion create a mesonic state (though it is an eigenvalue neither of Q do not spoil the degeneracy at large acting on the vacuum in eigenstates of the Hamiltonian (see a M B Q The strategy is the following. We consider infinitely heavy q It is well known that at high angular momentum (but below the s Assuming then heavy quarks of the fermionic string: and f semiclassical quark-quark pair, connected byRegge a trajectory fermionic s for baryonsσ made of light quarks, and the sl In a theory with Ψthought fermions, as a a fermionic string chromoelectric is fluxbaryons present with of as a the w Ψ kind fermion generated smeared by in the it operator We want to show that in the large- We assume the same path connecting in the fundamental representation are quenched in the large way the equality of the Regge slopes. definition of these states). At large separation in this sector toquarks get the degeneracy of the bosonic and fermio ergy), the mesons arepair, effectively connected described by by a arelated bosonic relativisti to string. the tension The Regge slope of meson are proportional to tensions do not depend on the mass of the quarks to probe the string states.the Even Regge if mesons trajectory and in baryons made the of form ( point along this path. At any be equivalently replaced bybaryonic one number more defined quark abovegeneralization in is of nothing the baryons but fundamenta was firstly the proposed usual by baryon Corrigan a M on the spin index. Assigning to is its spin index). We define operator nor of the angulartwo momentum), quarks while the operator mesons and baryons in QCD. Indeed since for SU(3) JHEP07(2009)073 b/f i . (2.9) (2.6) (2.8) ) the ) (2.14) (2.12) (2.13) (2.10) (2.11) 1 n, r | x 2.1 ( i Q i l 2

d~y b ], in the Hamilto- ) i y 16 ( the spectrum of the ~ argue, however, that A n, r | for more details): 1 is an index running on α N x harges explicitly breaks orm [ z C S ce describing the system r Z

erate. but is proportional to the ′ ould not aﬀect its tension. d-points will not spoil the i rge- quarks by heavy anti-quarks . The set of states , r ′ n ndix ) (2.7) ectrum of the eﬀective strings n 0 αβ

Q . ] R f Pexp a ν ( t

† f ′ ) we get: O , γ ) i ,r Q µ z ′ + a ( γ X [ λ 2.12 f n, r α,n l kβ | 0 i a µν λ = respectively as the spectra of the bosonic a . Since for SU(3) this theory is ordinary F SYM x γ 1 αnr b αβ λ n, r ) 1 4 } 3 | i 1 x k H ~γ B f n d γ − x k j R σ ( – 4 – f i n { n, r Z nr = = 4 | = 1 super Yang-Mills. We replace in ( σ X and Q ] = ] = 0 α d~y α i j a k ) 0 a β = = S ]= iα N S y and x α α ( i f ) , λ is the supercharge in the 4-component spinor notation) ) i 0 } ~ † ,Q d~y † A α | S, A ) b n [ α SYM α , from equation ( S 0 z S S α y b ( σ S S x ( ( i [ n, r | { B | Z ~ A α i S,H X 1 [ H stand for bosonic or fermionic and n, r n, r 0 x | h x b Z i b/f Pexp α X ) 0 = x Pexp ( R j b n ¯ ) Q σ 0 4 x ( d~z j ¯ Z Q = = α ˜ The result is not obvious because the presence of the static c If we consider SUSY transformations in the de Wit-Freedman f Chosen a bosonic state Let us discuss the relation with ˜ M B where the superscripts antisymmetric fermion by an adjoint fermion,where and necessary some heavy nian gauge and without squarks ( QCD-strings in the orientifoldin theory super coincides Yang-Mills, with which the we sp show to be boson/fermion degen SUSY, since we dothe spin not of introduce the heavyIn charges squarks particular, at as the having end-points well. onlyboson/fermion of quarks degeneracy. the We (and QCD-string will not sh squarks) at the en As we will see later on, planar equivalence implies that at la then the variation ofchromoelectric the charge density Hamiltonian of is the in heavy quarks general (see not appe zero, all the degenerate states (if any) in the same energy level are supposed to bein an presence orthonormal of basis two static of charges the in Hilbert subspa QCD, we will refer to the sets and fermionic QCD-strings. JHEP07(2009)073 . R (2.15) (2.16) (2.17) (2.18) (2.19) exist with , we expect α 1 anise the nu- x . b i . We assume that and with non-vanishing ) will vanish. The 1 . Another observa- n, r 0 | S n x ] between this pair of N Q nclude that the QCD- × 2.16 a ary terms and therefore result should be a good t 3 ,H † rge- α R of Polyakov loops. Despite nglet and therefore in this able to see how orientifold Q N S α = 0. to a linear dependence on ) nsion of the baryon. In other ia Polyakov loops will enable e, namely that the expectation a i . he points ) λ x 0 ( γ R ( P h . The formalism above translates in , √ and one spin component x . i n 3 . ) f d R i α y O ′ = ( ˜ . B ′ Z ′ , r P = ′ n

f n ′ = b n σ | i i , r ′ ) Re = ˜ n – 5 – M x n, r , the r.h.s. of equation ( b

n | ( creates a meson in the same energy level (at the , f σ α α P ˜ S 1 S M R h become degenerate. For this reason we expect that

′ Re → ∞ h = B , r creates a baryon in an energy level ′ b R i n α . 3

˜ B f and R are connected by a SUSY transformation n, r | ) ˜ α B M S

′ 2.13 and , r ′ ˜ n M

) with non-vanishing amplitude. f R ) ′ f n σ the energy levels are Fermi/Bose degenerate and we can reorg − are coordinates in b R n corrections will not be large and therefore the large- σ ( x,y /N The operators It is possible to re-formulate the above discussion in terms We consider the orientifold theory on the Euclidean space Consider the two-point function By copying the above result to the “orientifold theory” we co the following property: that at large charge separation, which means that at least one fermionic state Now we can compute the element matrix of the commutator [ Being the chromoelectric charge density localised only in t reason is that the chargeAs density a is result, not expected to give rise states, using equation ( At large meration of the fermionic levels requiring that mathematical terms the ideais that recovered SUSY in is the broken limit only of by large bound charge separation This implies that the operator the circle is large and thatvalue the of theory a is Polyakov in loop the confining that phas wraps the circle is zero where approximation for QCD. of the equivalence ofan the easier two connection formulations, with theplanar type derivation equivalence v can 0’ be string extended theory. in We the will string sector. also amplitude if and only if the operator the 1 tion is that forlimit SU(2) the the mass antisymmetric spectra fermion of becomes a si leading order in string tension of thewords meson is the identical Regge to slope the of QCD-string te mesons and baryons is identical at la JHEP07(2009)073 (2.23) (2.24) (2.21) (2.20) (2.22) ) is equal to . | . it belongs to y . 2.19 ns respect only a − x | SUSY f n i ) . riented (bosonic) strings. below. We identify the . y | βσ Q | ( 1 y y † its ends) with the meson − th ends) with the baryon P − − he orientifold planar equiva- es, and the invariance of the ) x x exp | x gure | ( b n b. f n f n open strings P A h above correlator ( βσ βσ − n = − ], and expanding the real part we get Q X 5 is saturated by oriented strings. These + exp exp | does not need to vanish. oriented b n . y f n Orienti f n A i A SUSY − ) σ i y ) n x n ( | X = y Orienti X † b ( n – 6 – i b n † ) P = = σ y ) P βσ ( x ) Q − ( is saturated by open QCD-strings with a heavy quark P x center symmetry, which implies that the correlator ( ) unoriented P h x Orienti P Orienti N exp ( i h i is saturated by open strings which contain heavy quarks ) + Z b n ) P y Orienti h y ( A i ( † = 1 Super Yang-Mills [ ) P y a. n P ( ) X N ) Orienti † Orienti x i i x ( ) P ) ( = y ) y ] hence P P ( ( h x Q h ( 17 P P ) P ) SUSY h x i x ( ) ( identically vanishes. In the orientifold theory the fermio a bosonic operator invariant under charge conjugation (i.e y P P ( h h † P ). P SUSY ) i ) gives x 2.1 ) = 1 SYM the correlator ( . The open QCD-string spectra of the orientifold theory. a. O y ( P N h 2.20 P ) The correlator In Being Re x = 1 SYM under the global global symmetry [ ( 2 P strings are both bosoniclence and ( fermionic. In the large N limit t Z Figure 1 and a heavy anti-quark at their ends, namely by oriented string (which containsstring a and quark the andstring unoriented an of string anti-quark ( (which at contain quarks at bo the same correlator in pure and finally supersymmetry implies that The correlator at both ends. Thus these string are We used the charge conjugationN invariance of both the theori h The string spectra of the orientifold theory is depicted in fi b. Un-oriented (fermionic) strings. the neutral sector), by orientifold planar equivalence the JHEP07(2009)073 ic (A.1) (which we ψ ]. (the Hamiltonian 12 er the classical field A i – edients are: A thank O. Aharony, O. 10 hat planar equivalence f the Hilbert space, by odel (type 0’ string the- ay that both the mesons string spectrum contains ds of open strings admit , s a bosonic oriented string ) e useful in this work, about zation and renormalization ted and the fermionic open y try” [ ons. − zation will be kept implicit. supersymmetric, the spectra of x ion which will enable to confirm ( 3 , and the matter field δ A i ij δ E AB δ = 1 super Yang-Mills contains stringy like 2 ig N of the gauge group). The canonical variables − – 7 – R = ) y corrections. Finally, it will be interesting to explore ( B j /N , A ) orientifold theory exhibits a degeneracy between the boson x ( A N i ) gauge theory with an antisymmetric (or symmetric) fermion E N ]. The SU( = 0 is assumed), the chromoelectric field 19 0 , A ) gauge theories in the Hamiltonian formalism. The main ingr 18 the Hilbert space defined as the space of the wave functions ov configurations; the Gauss constrain whichrequiring restricts invariance under the local physical gauge subspace transformations; o the Hamiltonian which defines the dynamics of the system. The canonical degrees of freedom are the spatial gauge field The above analysis is in perfect agreement with Sagnotti’s m N • • • will assume in thesatisfy generic the representation following (anti)commutation relationships: gauge All the formulae which follow make sense only once a regulari scheme is fixed. However for sake of simplicity, the regulari and the fermionic strings. ory) [ lives on a stackboth of bosons D-branes and in fermions. typestrings 0’ The are string bosonic unoriented. theory. open Althoughopen strings The this strings are string open exhibits orien theory a is Bose-Fermi non- degeneracy. 3 Conclusions In conclusion, we showed thatand we the can baryons generalize are QCD represented inand by such an the a open baryon w string. isthe The a meson same i Regge fermionic slope. un-orientedholds string. and Our that assumptions The the were color-singlet two mild: spectrum kin of we assumed t which means that the large- Acknowledgments A.A. is supported byBergman, the B. PPARC Lucini advanced and fellowship especially award. to We M. Shifman forA discussi Review of gaugeIn theories this in appendix we the want Hamiltonian toSU( review formalism the main concepts, which ar our result and tothe estimate relation the between our 1 approach and “effective supersymme objects. It will be interesting to perform a lattice simulat JHEP07(2009)073 ) x glet (B.1) (A.2) (A.3) (A.4) (A.7) (A.5) (A.6) is the covari- A i A A R iT x , 3 tter, coupled with two , + d ] i ω variant under local gauge ∂ ions, whose generators are: ψ is zero, and the dynamical al states of the whole system Ω[ ysical states; it is defined as: . = c heavy quark in the spatial R i uge transformations, and this ) representations, which we will D e: Q D x R i , i A . ( ) D x , T ψ x R ¯ y i 3 ψγ 3 A i R , discarding the non-physical states. d . − ¯ T ) Qγ Qd † + x . i βH x A ( ψ ( A − i 3 A T e βH ), as it were not coupled to any static i δ B ω = 0 ) component of the gauge field, while Ω( − ) A )+ i ab e x i ¯ x x δ Qγ ( 0 A.3 B ( ( 0 A i P O 2 . The interaction Hamiltonian is given by: a i A γ h 2 ) which plays the role of a Lagrangian multiplier g 1 phys A A – 8 – E G 2 x x i ψ = ( | Z D A + Z } A 2 ) i − ω = Tr 1 A i g G y − ( E = Z b is the operator representing the gauge transformation A ¯ i ψ )= int , E } ) x 2 x ( H exp x g 3 1 ( A 2 d a Z ) G ψ x { = ( Z A 0 ω P on the Hilbert space, and the Haar measure projects on the sin is the chromomagnetic field, and = ) } x A jk ( H A F A ω G A ijk ǫ R iT 2 1 i are the colour indexed in the representation {− = {− b A , i , and a static heavy antiquark in a 1 B x The Hamiltonian is invariant under local gauge transformat The Hamiltonian is given by: The partition function is given by the trace of ] = exp ω where ant derivative. charges. However the quarks modify the(heavy Gauss law. charges The plus physic dynamical degrees of freedom) must be in where The physical states are definedcondition to is be the invariant under Gauss local constrain ga without any external sourc sector evolves with the same Hamiltonian ( B Static heavy fundamentalIn charges this appendix westatic heavy consider charges a in the gaugecall fundamental theory respectively or antifundamental quark with and dynamicalpoint antiquark. ma We start with a stati but, since the heavy quarks are static, the current is the Polyakov loop. Finally the partition function is: This can be achieved by introducing the projector onto the ph where exp Ω[ states. In the path integral formalism, in the Hamiltonian formalism becomes the JHEP07(2009)073 f i (B.2) (B.3) (B.9) (B.5) (B.7) (B.6) (B.4) (B.8) n, r | . The | 2 x . − ] . 1 ω x | ) Ω[ ) is proportional ) | 2 D R = ( † x ) f 2 n,r R − ons, it can be diago- x ntation, for gauge in- 1 erted in the flux tube βV . . x l matter in the adjoint , and as the fundamental i − | † 1 i e ) x β, umber of gluinos gives rise 2 tion of the gauge theory in )tr Ω( ) with respect this basis of ( βH stance 1 x sector must transform as the n,r ¯ X Q rted in the flux tube; an even − x of the characters with respect stance behaviour: es. The projector on the space e Q B.4 + ) . Z , s. This can be easily understood 2 ) . ons in , b f )tr Ω( R ) ) tr Ω( i i ( ,x 1 0 0 1 x b R R n,r x x ( ( ( n, r n, r ) in the appropriate representations: | | 3 ¯ βV Q ); and the fermionic eigenstates x O O ) ) d . We will refer to these, as the states of − Q ) tr Ω( 2 R e R R h + + ( x P x ( ( ) ( h b β R R A b f n,r n,r ( n,r n,r X f b n n V ω V V σ σ Z ) ( – 9 – x ) ( = = β ) = Tr A )= b ( | f ) = ) = | i i 2 G 2 1 R R x x ( ( SYM Z n, r n, r − | f b − | i n,r n,r Z 1 1 V V H x H x | = | β, exp β, ( have eigenvalue ( ¯ Q ¯ SYM Q Z b i Q i Q † ) Z ): Z 2 )= R n, r x 2 | ( ,x f n,r 1 V x )tr Ω( ( 1 ¯ Q x Q P tr Ω( h In the path integral formalism, the partition function In the case of dynamical matter in the antisymmetric represe Since the Hamiltonian commutes with the gauge transformati representation under gauge transformations in the effective oriented string connectingof the the two oriented heavy string charg states is: antifundamental representation under gauge transformati transformation. This means that the states of the dynamical states we get for the correlators of Polyakov loops: The last equation essentiallyto comes the from the integration orthogonality withpresence the of Haar the measure. two heavy The charges partition is func therefore given by: variance only an even number of dynamical fermions can be ins bosonic eigenstates to the expectation value of two Polyakov loops Ω( have eigenvalue The eigenvalues are organized with respect to their large di nalized on therepresentation, space there are of both the bosonicconsidering and string that fermionic an states. state arbitrary numbersnumber In of of gluinos case gluinos can gives of beto rise inse dynamica fermionic to states. bosonic states, The while an eigenvalues odd are n functions of the di By expanding the trace of the partition function in eq. ( JHEP07(2009)073 , 1 x (C.4) (C.3) (C.5) (C.6) (C.2) (C.1) (B.10) (B.12) (B.11) entifold -component tifold theory is the charge . ] C ω . ) . x , Ω[ R ) 3 ( D R d ) ( where f 2 n,r zed to the case of a Ma- b t be inserted in the flux x n,r A A β l) de Wit-Freedman form: βV λ ¯ λ . βV − i , . ) e − ) D αβ x e i x . y ( )tr Ω( αβ C 3 γ ] on as above, we get: n,r 1 λ X d − ν A n,r x A X ecting a static heavy quark in = ¯ λ x A , γ T ) 2 ( n,r X † λ µ A α 3 i g β λ δ ( ) γ λ 2

tr Ω( 2 k β )[ ( g 1 1 γ , + AB x 2 5 ( x δ ) A γ i 1 3 0 x Orienti A µν d A ( γ k B ) Z 2 F )+ A Orienti A B i x g x 1 4 λ ( Z ( i = B + A a i − γ = 2 – 10 – k ω = g 1 E } ) γ i 2 = = ) x A k y D ( } ( + Orienti ) ) 2 . We will refer to these as the states of the unoriented A i 1 iE B x g 2 x † A i ( Orienti ¯ G λ ( ) x i , E 2 A i A β ) ) x Z 2 Z A i )= x i , λ x ( E x α = ( S, A A 2 S A g λ 1 S { 2 { )tr Ω( G 1 exp )tr Ω( x 1 x Z pair. This means that the unoriented string states in the ori Z = tr Ω( in the adjoint representation of the gauge group. Using four )= h QQ tr Ω( 2 pair. This means that the oriented string states in the orien H h A ¯ ,x λ Q 1 Q x ( QQ P The supersymmetric charge can be introduced in the (on-shel are purely bosonic. connecting a In terms of the elementary fields the supercharge is: However in the orientifold theory, string states exist conn C SYM in theIn Hamiltonian the formalism case of SYM,jorana fermion the formulae of appendix A must be generali and the generator of the local gauge transformation is: and another static heavy quark in spinors, the Majorana fermion obeys the constrain the SYM Hamiltonian is: For gauge invariance antube odd to number connect of a dynamicaltheory fermions are mus purely fermionic. Following the same constructi conjugation matrix. With the normalization given by: string. The projector onto the unoriented string states is: JHEP07(2009)073 ]. (C.7) (C.8) (C.9) N (C.10) ]. ]. ]. , SPIRES ] [ ]. SPIRES SPIRES : [ i ] [ SPIRES P ]. ] [ x . 3 , , ta d e theory is properly SPIRES [ miltonian as expected A : G SPIRES elationship gets an extra d. A k miltonian, but: ] [ A hep-th/0309013 [ nly if a regularization scheme Z ]. ]. ]. hep-th/0311190 [ , + arXiv:0901.3796 hep-th/0302163 [ , k x . k P 3 , Π d are identical thanks to the Gauss law. Regge trajectories revisited in the SPIRES SPIRES = k hep-ph/0409168 i SPIRES [ A γ , ] [ ] [ (2004) 384 P x (2004) 3 orientifold equivalence = 0 G 3 − i d A hep-th/0307097 N [ λ , is invariant under spatial translations and H and (2003) 170 0 β 0 Exact results in non-supersymmetric large- Refining the proof of planar equivalence SUSY relics in one-flavor QCD from a new 1/N QCD quark condensate from SUSY and the From super-Yang-Mills theory to QCD: planar i phys γ ¯ A γ S B 579 B 682 ψ λ | – 11 – Z αβ k ] ]. C D B 667 = 2 0 ]= = γ } S,H [ ¯ A S α hep-th/0412203 hep-th/0608180 hep-th/0403071 [ [ ¯ λ S , (2003) 191601 S,H SPIRES i S, 2 Phys. Lett. [ [ Nucl. Phys. { , , 91 + (In)validity of large- A j Nucl. Phys. Algebraic treatment of effective supersymmetry B , A i E expansion (1985) 201 (2005) 045015 (2006) 105019 kij N ǫ Skyrmions in orientifold and adjoint QCD A 86 Diquarks as inspiration and as objects Z D 71 D 74 Phys. Rev. Lett. , = k are the gauge invariant generalization of the spatial momen Π k ¨ Unsal and L.G. Yaffe, gauge/string correspondence orientifold field theories Nuovo Cim. Phys. Rev. Phys. Rev. expansion orientifold large- equivalence and its implications On the physical states, the supercharge commutes with the Ha Moreover the anticommutator of two supercharges is given by [1] L.A. Pando Zayas, J. Sonnenschein and D. Vaman, [2] F. Wilczek, [3] A. Armoni, M. Shifman and G. Veneziano, [4] A. Armoni, M. Shifman and G. Veneziano, [5] M. [6] A. Armoni, M. Shifman and G. Veneziano, [7] A. Armoni, M. Shifman and G. Veneziano, [8] A. Armoni, M. Shifman and G. Veneziano, [9] S. Bolognesi, [10] S. Catto and F. Gursey, References It obeys the Majorana condition Again, on the physical states the operators Π It is worthregularized to and remind renormalized. at Thewhich equation this preserves above point is SUSY valid that isSUSY o violating we chosen, term, are otherwise which the assuming must vanish commutation that as r the th cutoffthanks is to remove the Gauss constrain: local gauge transformations. It does not commute with the Ha while this is not true for instance on the string states. where Π JHEP07(2009)073 ] , egge 56B ]. ]. , ]. SPIRES [ SPIRES hep-ph/0608004 [ [ ]. SPIRES ] [ SPIRES ] [ (2007) 185 Nucl. Phys. Proc. Suppl. , hep-th/9509080 hep-ph/9912280 , B 650 , ]. ]. hep-th/0201169 [ hep-ph/0511143 [ SPIRES – 12 – Quark condensate in massless QCD from planar ] [ SPIRES Phys. Lett. [ ]. , ]. 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