20 May 1999

Ž .Physics Letters B 454 1999 207–212

Fundamental strings as black bodies

D. Amati a,b,1, J.G. Russo c,2

a ( )International School for AdÕanced Studies SISSA , I-34013 Trieste, Italyb INFN, Sezione di Trieste, Italy

c Departamento de Fısica, UniÕersidad de Buenos Aires, Ciudad UniÕersitaria, Pabellon I, 1428 Buenos Aires, Brazil´

Received 28 January 1999Editor: L. Alvarez-Gaume

Abstract

We show that the decay spectrum of massive excitations in perturbative string theories is thermal when averaged over theŽ .many initial degenerate states. We first compute the inclusive photon spectrum for open strings at the tree level showingthat a black body spectrum with the Hagedorn temperature emerges in the averaging. A similar calculation for a massiveclosed string state with winding and Kaluza-Klein charges shows that the emitted graviton spectrum is thermal with a‘‘grey-body’’ factor, which approaches one near extremality. These results uncover a simple physical meaning of theHagedorn temperature and provide an explicit microscopic derivation of the black body spectrum from a unitary S matrix.q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

High string excitations have a large degeneracy;the number of string states of a given mass increasesexponentially in the excitation mass measured in

Ž .string units string tension or inverse string length .This degeneracy is consistent with the unitarity ofthe perturbative S matrix. Unitarity guarantees thatthe analysis of final states that arise by the decay ofa massive state must allow to retrieve the informa-tion on how the original massive state has beenformed, i.e., on which is the coherent superpositionof the many degenerate microstates that have beenformed and subsequently decayed into the final statesunder analysis. A consistent and complete way ofanalysing final states is through semi-inclusive quan-

1 E-mail: amati@sissa.it2 E-mail: russo@df.uba.ar

tities, such as spectra, two particle correlations, threeparticle correlations, etc. The total set of these quan-tities is complete in the sense that it characterizes theinitial state or, equivalently, the coherent superposi-tion that created the final quantities in question.Semi-inclusive quantities are calculable in perturba-

Žtive string theory order by order in the string cou-. w xpling with a well-defined algorithm 1 , which was

extensively used in the past when string theory wasbeing applied to strong interactions.

In this paper a remarkable property of the decayspectrum of a massive string state will be found:when averaging over all string excitations of mass MŽ 2 .M 4 string tension , the decay spectrum exactlyaverages to a black body spectrum with Hagedorntemperature T . That the average procedure shouldH

wash out all the information stored in spectra andcorrelations leading to a total informationless ther-

w xmal distribution was conjectured 2 in order to

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00375-5

( )D. Amati, J.G. RussorPhysics Letters B 454 1999 207–212208

understand recently uncovered connections betweenstring the ory and black holes. For completeness wewill briefly mention in Section 4 the possible rele-vance for black hole physics of this non-trivial prop-erty of a unitary microscopic quantum theory gener-ating thermal properties in a decoherence procedure.The present results also unveil an important physicalmeaning of the Hagedorn temperature, as the radia-

Ž .tion temperature of a macroscopic averaged string.

2. Decay rates in open string theory

The inclusive spectrum for the decay of a particu-< : Ž .lar massive state F of momentum p s M,0 intom

Ž 2 .a particle with momentum k k s0 is representedm

by the modulus squared of the amplitude for the< : < : < :decay of F into a state F m k , summed overX m

< : < :all F . In string theory, the initial state F willX

be described by a specific excitation of the stringŽoscillators for simplicity, we restrict the discussion

.to bosonic string theory . If N is the occupationnŽ .number of the n-th mode with frequency n , a state

< :F of mass M will be characterized by a partition� 4 `N of NsÝ nN , withn ns1 n

`

†ˆ ˆ< : < :N F sN F , Ns na Pa ,Ý�N 4 �N 4 n nn nns1

X 2 m n † mna M sNy1, a ,a sd h , 2.1Ž .n m nm

w xwhere m,ns1, . . . , D. For large N, there are 3,4

Dq 1 Dy2'y a NNN sconst. N e , as2p ,4 (N 62.2Ž .

possible partitions, thus representing the degeneracyof the level N associated with excitations of massM. The total momentum pX of F will be pX spm X m m

< :yk . At the tree level, F will also be a superpo-m Xˆ < :sition of string excitations, satisfying N F sX

X < :N F ,X

X X X 2 X(N sa p q1sNy2k a Ny1 , 2.3Ž . Ž .0

Let us momentarily ignore twists or non-planar ef-w xfects of open string theories 3,4 . They will be

incorporated later. The inclusive photon spectrum forthe decay of a state F is then given by�N 4n

<² < < : < 2dG k s F V k FŽ . Ž .ÝF 0 X �N 4�N 4 nnX<F NX

=Vol S Dy 2 k Dy 3dk ,Ž . 0 0

Dy 1

22pDy 2Vol S s , 2.4Ž . Ž .Dy1

G ž /2

where Ý means the sum over all states FXF < XX N

Ž . Ž .satisfying the mass condition 2.3 , and V k is thephoton vertex operator

V k sj E X m 0 eik . X Ž0. , j k m s0, k 2 s0,Ž . Ž .m t m

` 1m m m †mX 0 sx q i a ya ,Ž . ˆ Ž .Ý n n'nns1

`

m m †m m'E X 0 sp q n a qa ,Ž . ˆ Ž .Ýt n nns1

m mx , p s id , 2.5Ž .ˆ n n

where j is the photon polarization vector, and wem

have set aX s1r2. The d-function of energy-

momentum conservation, coming from the zero-modepart of the amplitude, has been factored out from

ˆŽ .2.4 . Let us define a projector P over states ofN

level N as

dzNyNˆ ˆ < : < :X X XP s z , P F sd F , 2.6Ž .EN N N NN NzC

where C is a small contour around the origin. Intro-ˆ Ž .Xducing P in 2.4 we can convert the sum into aN

< :sum over all physical states F , and use complete-X

ness to write

dzXX ˆXyN X N² < < :F s z F V yk z V k F ,Ž . Ž .E XF �N 4 �N 4�N 4 n nn zC

2.7Ž .

3yD dGk F�N 40 nF k ' . 2.8Ž . Ž .F Dy2�N 4n dkVol SŽ . 0

( )D. Amati, J.G. RussorPhysics Letters B 454 1999 207–212 209

Let us now introduce a coherent set basis,`

†< : < :� 4l s exp l Pa 0; p , 2.9Ž .Łn n n mns1

Ž .where l , ns1, . . . `, is a set of arbitrary complexn

parameters, and define

dz dzXX

XyNyN)F s z zE E Xl l

Xz zC C

=ˆ ˆX N N² < < :� 4 � 4l W yk z W k z l ,Ž . Ž .n n

2.10Ž .

Ž . w m x w xwhere W k 'exp jPE X exp ik. X . From stan-t

w xdard properties of coherent states 3,4 , we find

dz dzXX

XyNyN pPŽ j qj .1 2)F s z z eE E Xl l

Xz zC C

=`

X n) nexp K ql Pl z zŁ n n n

ns1

) )ql PJ ql PJ , 2.11Ž .n n n n

1X n X nm m m m'J s n j qj z y k 1yz ,Ž .Ž .n 2 1 'n

1X n X n

) m m n m n m n'J s n j z z qj z y k z 1yz ,Ž .Ž .n 2 1 'n

K sj Pj nzX n . 2.12Ž .n 1 2

Ž .The spectrum F k may be directly computedF�N 4nŽ .from 2.7 using standard operator techniques orŽ . Ž . Žfrom the explicit expression 2.11 , 2.12 , and we

.omit Lorentz indices

N Nn n` E E)F k s F ,Ž . ŁF l l

)�N 4n ž / ž / linear in j jEl El 1 2ns1 n n)l sl s0n n

2.13Ž .

Ž .which follows after inverting 2.9 and noticing thatŽ . Ž .V k is represented by the linear term in j of W k .

< : Ž .For every F , the F k are easily obtained�N 4 Fn �N 4nand are polynomials in k .m

The spectrum obtained by averaging over stringstates with mass M is

1F k s F k . 2.14Ž . Ž . Ž .ÝN 0 F 0�N 4nNNN <F N

Ž .One can introduce again the projector 2.6 to con-vert the above sum into a sum over all states, so that

1 dz dzXX

XyNyNF k s z zŽ . E E XN 0XNN z zC CN

=ˆˆ X NNTr z V yk z V k . 2.15Ž . Ž . Ž .

Ž . Ž .The trace in 2.15 can then be computed from 2.11by integrating over l ,l) ,n n

1)m ) m yl Pln n

)F k s dl dl e F ,Ž . ŁHN 0 n n l lNN n ,mN j j1 2

2.16Ž .

<where means the term linear in j ,j , andj j 1m 2n1 2

Ž .then setting j sj . Using 2.11 we get1m 2 m

e pPŽ j 1qj 2 .

F k sŽ .N 0NNN

=dw dÕ X2yDyN NyNw f w ÕŽ .E Ew ÕC C1 2

=y1

) nexp J PJ 1yw qK ,Ž .n n n j j1 2

`X Xnf w s 1yw , wszz , Õsz . 2.17Ž . Ž . Ž .Ł

ns1

C and C are two small contours around zero. The1 22Ž .extra factor f w in the integrand arises as usual by

Žincorporating the ghost contribution the Dy2power corresponds to the Dy2 transverse coordi-

.nates . Thus we obtain

1F k s j jŽ .N 0 m n

NNN

=dw dÕ X2yDyN NyNw f w ÕŽ .E Ew ÕC C1 2

= p p qh V Õ ,w , 2.18Ž . Ž .Ž .m n mn

` n n ynw Õ qÕŽ .nV Õ ,w s n Õ q . 2.19Ž . Ž .Ý nž /1ywns1

( )D. Amati, J.G. RussorPhysics Letters B 454 1999 207–212210

Ž X .The integral over Õ gives N -N

X 2yDX2 yNj NyN dw w f wŽ . Ž .F k s .XŽ . EN 0 NyNNN w 1ywN

2.20Ž .X Ž .For large N , the integral in 2.20 can be computed

by a saddle point approximation, with the maincontribution coming from w;1. This is similar tothe familiar calculation of the number of states NNNŽ . Ž .2.2 of level N. The behavior of f w at w(1 iswell known and given by

p 2y1r2 yf w (const. 1yw e . 2.21Ž .Ž . Ž . Ž .6 1yw

X'Therefore the saddle point is at lnw(yar2 N ,Ž . Xwhere a was defined in Eq. 2.2 . Using N sNy

X X'2k a N we obtain for large N0

' ''a Ny2 k a N ya N0 XeX2 'F k (j 2k a N .Ž .N 0 0 yak a'0 X1ye

2.22Ž .Dq1X

4Ž .We have set NrN (1, since in the regime ofX Žinterest, k <M, NrN (1 the emission of a pho-0

ton with energy k of order M and higher is sup-0X'y N '.pressed by a factor of order e . Thus N

X' 'y N (yk a and0

ya k a'0 XeX2F k (j 2a k M , 2.23Ž . Ž .N 0 0 yak a'0 X1ye

or

k 0ye TH Dy 2dG k (const. k dk , 2.24Ž . Ž .N 0 0 0k 0

y1ye TH

1where T is the Hagedorn temperature T s .H HXa a'

ŽThus the radiation spectrum from a macroscopic i.e..averaged over all degenerate microscopic states

Žstring of mass M is exactly thermal despite thespectrum for each microstate being absolutely non-

.thermal . The way the exact Planck formula arises inthe final result is striking, since the exponents in thenumerator and denominator have different origins.

Let us now show that the non-planar contributiondoes not change the above result. It arises in open

string theory due to the fact that the photon may beemitted by any of the two ends of the string. So,strictly speaking, we should have written

dG kŽ .F 0�N 4n

21

< <s F V k qQ V k Q FŽ . Ž .Ž .Ý X �N 4¦ ;n'2X<F NX

=Vol S Dy 2 k Dy 3dk , 2.25Ž . Ž .0 0

Ž .at the place of 2.4 , where Q is the twist operatorw x Ž .3 . The modulus squared involving V k – and that

Ž .with Q V k Q – leads each to half of the planarŽ .result 2.24 obtained before. The cross product gives

rise to the non-planar contribution that may be com-puted analogously to the planar loop diagram, ofwhich our spectrum represents the absorptive part.

Ž .Instead of 2.15 for the non-planar contribution onehas

1 dz dzXX

XyNyNF k s z zŽ . E En .p . XN 0XNN z zC CN

=ˆˆ X NNTr z Q V yk Q z V k .Ž . Ž .

2.26Ž .w xUsing the explicit expression for Q 3 and comput-

w xing the trace 5 one finally obtains

F kŽ . n .p .N 0

1s j jm n

NNN

=dw dzX

X2yD X NyNyNw 1yw f w zŽ . Ž .E E XXw zC C1

= p p qh V Õ ,w , 2.27Ž . Ž .Ž .m n mn

wy zX

Ž . Xwhere V is given by Eq. 2.19 with Õs . The1y z

contour integral in zX now selects the pole at zX sw.The remaining integral over w is dominated by thesame saddle point that allowed the evaluation ofŽ .2.20 . The final result is

2j aya k a'0 XF k ( e 2.28Ž . Ž .n .p .N 0 '2 N

which is smaller by a factor -1rN with respect toŽ . Ž .the planar result 2.23 or 2.24 that dominates the

Ž .sum 2.25 .

( )D. Amati, J.G. RussorPhysics Letters B 454 1999 207–212 211

3. Spectrum of a charged closed string state

Let us now consider closed bosonic string theorywith one dimension compactified on a circle ofradius R. Let m and w be the integers representing0 0

the Kaluza-Klein momentum and winding number ofŽthe string state along the circle we will assume

.m ,w )0 . The mass spectrum is given by0 0

X 2 ˆ X 2 ˆ X 2a M s4 N y1 qa Q s4 N y1 qa Q ,Ž . Ž .R q L y

3.1Ž .

m w R0 0where Q s " , Consider the decay rate of a" XaR

state of level N , N of given charges m ,w , withL R 0 0

N yN sm w , by averaging over all physicalL R 0 0

states of charge m ,w and level N , N . The calcu-0 0 L R

lation factorizes in a left and right part, and it issimilar as in the previous section. We assume thatN X and N X are sufficiently large for the saddle-pointR L

approximation to apply. The final result is

k k0 0y ye eT TL R2 2dG k sconst. j MŽ .N N 0 k kR L 0 0

y y1ye 1yeT TL R

=Vol S Dy 2 k Dy 1dk , 3.2Ž . Ž .0 0

2 2 2 2( (2 M yQ 2 M yQq yT s , T s . 3.3Ž .R LX X' 'a a M a a M

Ž .Eq. 3.2 is remarkably similar to the analogousw x Ž w x.result derived for D-branes in 6 see also 7 ,

despite the two calculations being rather different,and describing different physical systems. Indeed,the D-brane calculation involves only a single oscil-lator a,a† that excites modes with one unit of

w xKaluza-Klein momentum 6 ; the present elementarystring calculation involves all excitations a ,a† , nsn n

1, . . . ,`, in a very specific way dictated by the vertexinsertion.

w x Ž .By analogy with black holes 7 , Eq. 3.2 may bewritten as

k 0ye T

Dy2dG k sconst. s k k dk ,Ž . Ž .N N 0 0 0 0kR L 0y1ye T

3.4Ž .

y1 y1 y1 Ž .where T sT qT and s k is the ‘‘greyR L 0

body’’ factor

k 0y1ye T

s k sk . 3.5Ž . Ž .0 0 k k0 0y y1ye 1yeT Tž / ž /L R

The radiation vanishes if the initial string statesaturates the BPS condition MGQ , i.e. when MsqQ . For a near BPS state, M(Q , N 4N , oneq q L R

gets

2 2(M yQ2 qT(T s . 3.6Ž .R X' Ma a

In the superstring theory, owing to the supersymme-try of the BPS state, corrections to the ‘‘free string’’picture are expected to be smaller for near-BPSconfigurations, which might allow to extrapolate Eq.Ž .3.6 from the weak to the strong coupling regimew x8 .

4. Discussion

The appearance of the Hagedorn temperaturecharacterizing the radiation in the decay of a massivemacrostate in string theory at zero temperature shedslight on an old problem concerning the physicalinterpretation of the Hagedorn temperature. TheHagedorn temperature was traditionally interpretedas the temperature at which the canonical thermalensemble of a string gas breaks down. The technicalreason is simple – at higher temperatures the integraldefining the thermal partition function diverges –and it is related to the fast growing of the leveldensity with mass. The question is what is the physi-cal picture behind this instability. The present results

Žlead to the following interpretation this seems con-w x.sistent with other suggestions 9,10 . When an open

string in a macrostate with N41 is placed in athermal bath of temperature T -T , the stringbath H

Ž .will decay by emitting massless as well as massive

( )D. Amati, J.G. RussorPhysics Letters B 454 1999 207–212212

particles, as a black body of temperature T , untilH

the level decreases to a point where the saddle-pointprediction T sT ceases to apply, and the realstring H

temperature of the string is of order T . A situationbath

of equilibrium occurs immediately if one places thestring in a thermal bath with T sT . When Tbath H bath

)T , the energy of the string increase endlessly,H

since it can only emit at a fixed rate determined bythe temperature T . The system becomes highlyH

unstable, and there cannot be any thermal equilib-rium. The canonical thermal ensemble must thereforebreak down precisely at TsT .H

It is interesting to witness how these statistical orclassical concepts stem from microstate computa-

Ž .tions through average procedures decoherence . Thestring scale is of course present in the spectrum andin the correlations of single microstates, but it is onlyafter averaging that appears as the temperature char-acterizing the emission. The emergence of classicalproperties in a computable weak coupling stringregime was indeed one of the motivation for this

w xinvestigation. It substantiates the conjecture 2 that asimilar mechanism is at work at large couplings inorder to generate from string theory classical con-cepts such as space-time geometry, causal relationsand black hole physics.

Finally, it would be interesting to investigate pos-sible cosmological implications related to the decay

w xspectrum and lifetime of cosmic strings 9,11 .

Acknowledgements

J.R. would like to thank SISSA for hospitality.This work was partially supported by EC ContractERBFMRXCT960090, by the research grant on The-oretical Physics of Fundamental Interactions of theItalian Ministry for Universities and Scientific Re-search, and by Conicet.

References

w x Ž .1 A.H. Mueller, Phys. Rev. D2 1970 2963.w x2 D. Amati, Black holes, string theory and quantum coherence,

hep-thr9706157.w x3 V. Alessandrini, D. Amati, M. Le Bellac, D. Olive, Phys.

Ž .Reports C 1 1971 269.w x4 M. Green, J. Schwarz, E. Witten, Superstring Theory, vols.

IrII, Cambridge, 1987.w x5 D. Gross, A. Neveu, J. Sherk, J. Schwarz, Phys. Rev. D 2

Ž .1970 697.w x Ž .6 C. Callan, J. Maldacena, Nucl. Phys. B 472 1996 591.w x Ž .7 J.M. Maldacena, A. Strominger, Phys. Rev. D 55 1997 861.w x Ž .8 A. Sen, Mod. Phys. Lett. A 10 1995 2081.w x Ž .9 D. Mitchell, N. Turok, Phys. Rev. Lett. 58 1987 1577.

w x Ž .10 M. Bowick, S. Giddings, Nucl. Phys. B 325 1989 631; S.Ž .Giddings, Phys. Lett. B 226 1989 55; D. Mitchell, B.

Ž .Sundborg, N. Turok, Nucl. Phys. B 335 1990 621; N. Deo,Ž .S. Jain, C.I. Tan, Phys. Rev. D 40 1989 2626; M. Laucelli

Meana, M. Osorio, J. Puente Penalba, Phys. Lett. B 400˜Ž .1997 275.

w x Ž .11 A. Vilenkin, Phys. Rep. 121 1985 263.