FINE STRUCTURE OF STRING SPECTRUM IN ADS$ X S5

ZAREMBO K.; ZIEME S.

Pis'ma v ZhETF, vol.95, iss.5, pp. 243-247

Fine Structure of String Spectrum in AdS$ X S5

K. Zarembo, S. Zieme1^ Nordic Institute for Theoretical Physics, SE-106 91 Stockholm, Sweden

Submitted 23 January 2012

The spectrum of an infinite spinning string in AdS§ does not precisely match the spectrum of dual gauge theory operators, interpolated to the strong coupling regime with the help of Bethe-ansatz equations. We show that the mismatch is due to interactions in the string cr-model which cannot be neglected even at asymptotically large't Hooft coupling.

According to the AdS/CFT correspondence [1], N = = 4 supersymmetric Yang-Mills theory (SYM) and string theory in AdS5 x S5 have a common spectrum that continuously interpolates between the loop-corrected dimensional analysis at weak coupling and the string oscillator spectrum at strong coupling. The complete integrability of the AdS/CFT system makes the non-perturbative interpolation amenable to an exact description by methods of Bethe ansatz [2]. The string interpretation of the spectrum, however, is quite subtle and our goal is to find a potential resolution of these subtleties.

We shall concentrate on a specific set of states related to twist-two operators tr ZD^_Z. Here Z is a complex scalar field in SYM and D+ is the covariant derivative in light-cone direction. The twist operators constitute presumably the most studied sector of the SYM spectrum [3]. Their anomalous dimensions scale logarithmically with spin: Ag(A) — S ~ 2rcusp(A) InS, where the cusp anomalous dimension rcusp(A) is a non-trivial function of the 't Hooft coupling A = £yMJV, which can be computed non-perturbatively with the help of the Bethe-ansatz equations [4]. On the string side, twist operators are described by a string spinning in the Anti-de-Sitter space [5]. When the spin is very large, the string becomes essentially infinite, extending all the way to the boundary. The energy density of this long string is equal to the cusp anomalous dimensions rcusp(A).

We will be interested in the spectrum of small fluctuations on top of the long string, which are dual to operators with extra field insertions, schematically: t r Z D1.' »P1 D1': ... D1'; Z, where ^ can be a field strength, a fermion or a scalar. Each insertion corresponds to an elementary excitation above the ground state. The spectrum of elementary excitations can be found exactly [6, 7] by solving the Bethe-ansatz equations [4], and should agree at strong coupling with the spectrum of the string in light-cone gauge. A detailed comparison reveals, however, several mismatches [8].

Since the above operators have many uses, for instance they govern the collinear limits of scattering amplitudes [9], it is important to understand how these discrepancies are resolved.

The string oscillation modes in light-cone gauge are two-dimensional massive particles, whose interactions are suppressed by 1/y/X. Let us list the 8j + 8/ modes of the string in AdS$ x S5, together with their masses and the corresponding worldsheet fields [10, 11]:

AdS3 transverse (#) : to2 = 4, AdSs outside AdS3 (X,X*) : m2 = 2, S5 (Fa, 0= 1,... ,5) : to2 = 0, Fermions i = 1,... , 4) : to2 = I. (1)

The fermions form four 2d Dirac spinors with eight degrees of freedom on shell. The SYM-spectrum, continued to strong coupling, consists of [6]:

(Field strength)^ (2) : to2 = 2£2,

Scalars (6) : to2 = e ,

Fermions (8) : m2 = 1, (2)

where in brackets we indicated the number of degrees of freedom of each excitation. The agreement holds literally only for fermions. The heavy boson from AdS3 is absent in the exact spectrum. The S5 modes are massive, and there are six of them instead of five. A single insertion of the field strength (£ = 1) matches with the AdS$ modes X, X*. Multiple field strength insertions in addition can form bound states (Bethe strings) which are not directly visible in the string spectrum. The binding energy of the .¿-string, Efni(p) = £E1(p/£) - Et(p), is small at strong coupling [6]:

1) e-mail: zarembo0nordita.org

p*«(n) - (£2+p2)4 1 [P)~ 12A£6 (2£2 + p2) ' (6)

These mismatches indicate that the string spectrum is strongly affected by interactions, even at asymptotically large A. The mechanism for mass generation of

244

K. Zarembo, S. Zieme

the S5 modes, which is clearly non-perturbative, was understood in [12] and later confirmed by Bethe-ansatz calculations [13]. Since all other modes are massive, the low-energy effective theory at worldsheet momenta p -C 1 is the 0(6) cr-model, which is asymptotically free and generates a mass gap through dimensional transmutation. The first correction to the dispersion relation of the S5 modes that goes beyond the low-energy approximation was computed recently both in string theory [8] and from the Bethe-ansatz equations [6]. In both cases the result has the form:

\p)=p2

cX-1'2

pA + 0( A"1) (p»to), (4)

but the coefficient is different: cstrmg = 7/3 « 2.333 [8] while cBA = v/2r4(l/4)/(24 • 31//47r) « 2.463 [6]. This is quite surprising since the p4 term is relevant in the UV at p2 to2 when perturbation theory is supposed to work.

We thus shall address the following questions: 1) can the light AdS scalars form bound states? 2) what is the fate of the heavy AdS mode? 3) what accounts for the difference between perturbative and exact values of the p4 coefficient in the dispersion relation of the S5 modes? Our starting point is the string cr-model in light-cone gauge, with the action expanded to quartic order in the fields [11, 8]:

+ 4j#Fa(#II_pa6Vs\P - Vsf n+ + 6#(VsX*f n_p|f4 - VsX^'n+p6^) -- 2iYadtYb&70n+pa6^ + (#70n+pa6^)2 -

- (^7oII+<l>)2. (5)

The notations follow [11, 8], with the exception of fermions which we have brought into the 2d Dirac form. We use 7a = (—a1, a3) as the basis of 2d Dirac matrices, n± = (1 ± 7s)/2, V, ,% - 1, pa,p6 are the 4x4 chiral components of the 6d Dirac matrices, and pAB = pI^tpB]. The action is written in Euclidean signature and is multiplied by an overall factor of VX/in.

We begin with the bound states of the light scalars X. The small binding energy makes these bound states non-relativistic at strong coupling. The binding energy can

thus be derived from a low-energy effective Hamiltonian. The effective potential is determined by matching the XX XX scattering amplitude in the cr-model to the Born amplitude M(q) = ^2 (2V2)2 / dx e -iqxVeS(x), where q is the momentum transfer in the ¿-channel. The XX scattering at tree level proceeds through the exchange of the # boson. The t- and «-channel diagrams combine to

M =

16tt

o(q2)

VeS{x) =

Vx

S[x). (6)

The effective potential is thus an attractive delta function, which has one discrete level with energy 7r /2V2A. The solution of the Schrodinger equation for I particles interacting pairwise via the delta-function potential is also known [14]. There is a single bound state for each I with binding energy

^jbind _

1)

24A

(7)

that agrees nicely with (3) in the static regime.

To address the fate of the heavy bosonic mode we need to study quantum corrections to its propagator. Since the heavy boson is a factor of two heavier than the fermions, it may dissolve in the continuum of the two-particle states, as it happens in a similar context in AdS^ [15]. The pole of the boson propagator disappears by moving onto the unphysical sheet of the complex energy plane. Consider first one-loop corrections:

G-i(p) =p2+4*

C_

Vx

Vp2

(8)

where we assume that the momentum is very close to the threshold: p2 ^4 and keep only the most singular part of the polarization operator. The would-be pole lies at the edge of the two-particle continuum and may disappear, depending on the sign of C. If C is negative, the two terms in (8) have opposite signs for 0 > p2 > —4. They cancel at some p2 and the pole remains on the physical sheet just below the threshold. However, the explicit calculation [8] shows that C is positive. As we shall see, the positivity of C is a simple consequence of unitarity. The pole then disappears (it moves into the unphysical sheet), and the only remaining singularity of the Green's function is the two- particle cut. If (8) were exact, we would conclude that the heavy boson does not exist as an independent excitation. But there are other corrections to the boson propagator that should also be taken into account. First, the mass-shell conditions for the boson and fermions get loop corrections [8]. Unless ej(p) = 2e/(p/2), these corrections shift the pole away from the threshold, and then (8) will have an isolated

Fine Structure of String Spectrum in AdS$ x S5

245

zero. The one-loop correction to the boson dispersion relation is the constant part of the polarization operator, while the compensating correction to the fermion mass-shell condition, p2 = 1 + e^^/VX, affects the po-

larization operator at two loops:

Disc 11(2) (p) = C(2) \/p2 +4

D,

(2)

(9)

where D(2) must be equal to in order for the

second term to combine into the perturbative expansion of the square root in (8). But what if And also why D^ =0? If D ^ 0, the heavy particle would decay rather than dissolve in the continuum. The unexpected softening of the threshold singularity, in fact, can be traced to the structure of the boson-fermion vertex [8]: the threshold singularity of the polarization operator is related by unitarity to the <I> > lI"i' amplitude, Fig.a:

Possible unitarity cuts of the fermion loop in the polarization operator of the heavy boson

ImII(p) =

4p2 s/p2 + 4

(10)

where Ah.^ff(p) is the ### vertex with both fermions on-shell: Ah.^ff(p) = v(k)F(p;k,p — k)u(p — k). The fermion and antifermion wavefunc

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