Black Holes and the SYM Phase Diagram Miao Li University of Chicago

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1-1-1999 Black Holes and the SYM Phase Diagram Miao Li University of Chicago

Emil Martinec University of Chicago

Vatche Sahakian Harvey Mudd College

Recommended Citation Miao Li, Emil Martinec and Vatche Sahakian. "Black holes and the SYM phase diagram." Phys. Rev. D 59, 044035 (1999). doi: 10.1103/PhysRevD.59.044035

This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. PHYSICAL REVIEW D, VOLUME 59, 044035

Black holes and the SYM phase diagram

Miao Li,* Emil Martinec,† and Vatche Sahakian‡ Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637 ͑Received 17 September 1998; published 29 January 1999͒ Making combined use of the matrix and Maldacena conjectures, the relation between various thermodynamic transitions in super Yang-Mills ͑SYM͒ theory and supergravity is clarified. The thermodynamic phase diagram of an object in DLCQ M theory in four and five non-compact space dimensions is constructed; matrix strings, matrix black holes, and black p-branes are among the various phases. Critical manifolds are characterized by the principles of correspondence and longitudinal localization, and a triple point is identified. The microscopic dynamics of the matrix string near two of the transitions is studied; we identify a signature of black hole formation from SYM physics. ͓S0556-2821͑99͒03604-8͔

PACS number͑s͒: 04.70.Dy

I. INTRODUCTION AND SUMMARY otherwise the black hole fills the longitudinal direction and becomes a black string.1 We will call the transition at N The thermodynamic phase structure of a theory is an ex- ϳS a localization transition. Other geometrical, but non- cellent probe into the underlying physics. Transitions among longitudinal, effects of this sort may also be expected ͓6͔. different phases reflect the dynamics via the stability or If there existed a single framework—one description that metastability of various configurations, while order param- realizes the different phases of M theory—then this theory eters often characterize global properties. M or string theory should exhibit the critical phenomena associated with these is no exception in this regard. various transitions. The matrix theory conjecture proposes In particular, two thermodynamic transition mechanisms such a framework: U(N) SYM on a torus is supposed to in M or string theories have recently been a focus of the manifest a rich structure of M or string theory phases such as literature. One example occurs when the curvature near the black holes, strings and D-branes ͓7͔. SYM thermodynamics horizon of a supergravity solution becomes of the order of is then endowed with a cornucopia of critical behaviors. the string scale; the state becomes ‘‘stringy,’’ and acquires Field theoretically, transitions between SYM phases are pos- an alternative string theoretical description, either by a per- sible as functions of the size and shape of the torus, the YM turbative string or by supersymmetric Yang-Mills ͑SYM͒ coupling, the temperature, and the rank of the gauge group. D-brane dynamics ͓1͔. This metamorphosis might be re- A complementary recent conjecture of Maldacena ͓8,9͔ garded as a phase transition in the embedding theory, and is provides us with the tools to study SYM thermodynamics in known as the correspondence principle. A second transition regimes previously considered intractable. It states that the mechanism is associated with the mechanics of localizing a macroscopic physics of M or string theory in the vicinity of state in a compact direction. Of particular interest in discrete some large charge source ͑a regime accurately described by light-cone quantization ͑DLCQ͒ is the localization effect in supergravity͒, is equivalent to that of super Yang-Mills the longitudinal direction Rϩ ͓or R11 in the infinite momen- theory. Finite temperature SYM physics acquires in certain tum frame ͑IMF͔͒ ͓2,3,4,5͔. Particularly, a state with fixed regimes a geometrical description, that of the near horizon rest mass M and N units of DLCQ momentum satisfies the region of near-extremal supergravity solutions. Renormaliza- condition tion group ͑RG͒ flow is mapped onto transport in the geometry about the horizon; correlation functions in the SYM N probe different distances from the horizon as one changes the R Ͻ ϵqϪ1. ͑1͒ ϩ M separation of operator insertions relative to the correlation length ͑thermal wavelength͒ in the SYM. Our plan is to use the Maldacena conjecture, along with If the system characterizes an object of size r0 , then we need the interpretation of the SYM physics from the matrix theory r0ϽRϩ to localize the object. For example, for a black hole perspective, to piece together the phase diagram of an object satisfying the equation of state Mr0ϭS, we need in DLCQ M theory. In parallel, we will end up making statements about the critical behavior of SYM thermodynamics NϾS, ͑2͒ on the torus well into non-perturbative field theory regimes.

1This intuitive argument ignores the effects of gravity. The mo- *Email address: [email protected] mentum of an object back-reacts on the nearby geometry and in †Email address: [email protected] particular changes the available proper longitudinal volume; such ‡Email address: [email protected] effects do not affect the conclusion ͑2͒, however ͓5͔.

FIG. 2. The string self-interaction potential as a function of relative separation x along the string, for pϭ4,5. ‘‘coexistence phase’’ of a matrix string with SYM vapor. On FIG. 1. The proposed thermodynamic phase diagram for the p the upper left part, the system evaporates into a weakly ϩ1d SYM on the torus, or the DLCQ IIA theory, obtained by coupled SYM gas, over sufficiently short time scales that one tracking an object in matrix theory. On the horizontal axis is the IIA cannot think of the ensemble as that of a single object in string coupling, which is the aspect ratio of the SYM torus. The spacetime. There is a ‘‘triple point,’’ a thermodynamic criti- vertical axis is the density of states of the object. cal point of the DLCQ string theory where the three transition manifolds coincide. We focus on type IIA theory on T p with pϭ4,5 (4ϩ1d A brief description of the physics of the diagram is as and 5ϩ1d SYM theory͒. One reason for this is that gravita- follows: In type II DLCQ string theory on T p, with p tional interactions are longer-range, and sufficiently strong to ϭ4,5, there exists a ͑longitudinally wrapped͒ Dp-brane be important for state transitions, in lower dimensions phase; it is unstable at the Banks-Fischler-Klebanov- ͑higher p͒. Also, the cases with pϭ1 and pϭ2 require Susskind ͑BFKS͒ point ͑the horizontal line at NϳS in the slightly more effort; and the case pϭ3 is conformal—the diagram͒ to the formation of a black hole because of longi- SYM coupling gY is dimensionless, so that quantitative con- tudinal localization effects. Along another critical curve ͑the trol is required for some questions to be addressed. We con- diagonal line above NϳS͒, the Dp brane freezes its strongly fine this report to a more qualitative sketch of the phase coupled excitations onto a single direction of the torus, mak- diagram; for example, we ignore numerical coefficients in ing a transition to a perturbative string through the corre- state equations. We hope to return to a discussion of the spondence principle. In this regime, the thermodynamics is situation for pр3 elsewhere. In particular, the cases p that of a near-extremal fundamental ͑IIB͒ string supergravity ϭ3,2 are relevant to both matrix theory and string theory in solution, with curvature at the horizon becoming of order the anti–de Sitter space, and thus their phase diagrams should be string scale. The correspondence mechanism also applies on rather interesting. the other side of the BFKS transition; in this case, a matrix Our main conclusions come in two pieces. The first con- black hole makes a transition to a matrix string when it ac- sists of an overview of previous observations ͓1,3,5,9,6͔, put quires string scale curvature at the horizon. A coexistence in a new unifying perspective. Figure 1 summarizes the situ- phase, where both matrix string and SYM gas excitations ation. It is the phase diagram traced out by a single object in contribute strongly to the thermodynamics, may exist in the matrix theory on T4 or T5. We are assuming that the differ- region indicated on the diagram; this depends on the extent ent states we track are characterized by long enough life- to which the object persists long enough to treat it using the times so that it makes sense to describe them thermodynami- methods of equilibrium thermodynamics. cally, as ͑meta͒stable phases. In super Yang-Mills, one has in Our second set of results concerns the dynamics that leads mind starting the system with all scalar field vacuum expec- to the correspondence transition, and is summarized by Fig. tation values bounded in some appropriately small region, 2. The plot depicts the mutual gravitational interaction en- such that the interactions sustain a long-lived cohesive state. ergy between a pair of points on a typical ͑thermally excited͒ In the figure, the limit of validity of the SYM description macroscopic matrix string, as a function of the world-sheet for the DLCQ string theory is determined by the upper right distance x along the string separating the two points. This curve. In the shaded region, the theory is sufficiently strongly potential governs the dynamics of the matrix string near a coupled at the scale of the temperature that it is not accu- black hole or black brane transition, as it is approached from rately described by super Yang-Mills theory; rather, one the weak coupling side. A bump in the potential occurs at the must pass to the six-dimensional ͑2,0͒ theory ͓10͔ for p thermal wavelength N/S for pϭ4,5 ͑five or four noncompact ϭ4, or the ill-understood ‘‘little string’’ theory ͓11,12͔ for spatial directions͒; in these dimensions, the correspondence pϭ5. We will see that the dynamics of interest to us occurs transition to a black hole is indeed caused by the string’s outside this region. We identify several phases in SYM on self-interaction, as discussed in ͓13͔. For smaller p ͑more the torus; a black hole phase, a string phase, a phase of p noncompact spatial directions͒, there is no bump; similarly, ϩ1-dimensional strongly interacting SYM, and perhaps a in ͓13͔ the self-interactions could not cause a spontaneous

044035-2 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035 collapse to a black object. We will see that the height of the ͑i͒ This theory is equivalent to an (M,lpl) theory on the bump is proportional to the gravitational coupling, such that DLCQ background we denote by D1,1ϫT pϫR9Ϫp, where it ‘‘confines’’ excitations of the string on the strong-coupling D1,1 isa1ϩ1 dimensional subspace compactified on a light- p side of the correspondence transition. like circle of radius Rϩ , and the torus T has radii Ri (i This result supports a suggestion ͓14,15͔ to describe the ϭ1...p). The map between the two theories is given by black hole phase as clustered matrix SYM excitations of size 2 N/S. These correlated clusters were invoked in order that the lpl lpl object with NϾS can be localized in the longitudinal direc- xϭ , y iϭ , ͑4͒ Rϩ Ri tion. Such a localization necessarily involves the longitudinal momentum physics of matrix theory. We find that a plausible with N units of momentum along Rϩ . argument for the dynamics with this potential gives the pre- ͑ii͒ The dynamics of the M¯ theory in the above limit can viously identified correspondence curves as the boundaries be described by a subset of its degrees of freedom, that of N of validity of the matrix string phase. For NϽS, one finds D0 branes of the IIA theory, up to a certain UV cutoff. the transition to the interacting pϩ1 SYM phase shown in The two propositions above, in conjunction, are referred Fig. 1; while for NϾS, one finds the transition to the matrix to as the matrix conjecture ͓7,17͔. black hole phase. Accounting for the latter transition requires T-dualizing on the ¯R ’s, we describe the D0 brane physics taking into consideration longitudinal momentum transfer ef- i by the pϩ1d SYM of N Dp branes wrapped on the dualized fects as in ͓14͔; we justify this by a string theory amplitude torus. We remind the reader of the dictionary needed in this calculation involving winding number exchange in a dual process picture. We thus conclude that we have identified the char- acteristics of the microscopic mechanism of black hole for- ¯Rϭ¯g ¯l ¯l 3 ϭ¯g ¯l 3 , mation from the SYM point of view. sЈ s pl sЈ s The plan of the presentation is as follows: in Sec. II, we ¯ p ¯2 review the two conjectures ͑matrix and Maldacena͒ we will l s l s use in the analysis of the phase diagram. In Sec. III, we bring ¯gsϭ¯gsЈ ⌺iϭ . ͑5͒ ¯ ¯ together previous observations with some new ones to map ⌸Ri Ri out the phase diagram for the DLCQ matrix string. Section ¯ IV extends our arguments at the triple point to the cases of The first line is the MϪIIA relation, the second that of singly and doubly charged black holes. We also discuss a toy T-duality. The limit ͑3͒ then translates in the new variables mechanism for clustering of SYM excitations for the singly to charged case, at the BFKS point. Section V discusses the ¯␣Ј 0, with g2 ϭ͑2␲͒pϪ2¯g ¯␣Ј͑pϪ3͒/2 and ⌺ fixed, self-interaction of the matrix string, the identification of the → Y s i bump potential and comments about its dynamics. We out- ͑6͒ line in the Appendices the calculation of the potential, and a where the nomenclature g2 and ⌺ refers to the coupling and scattering amplitude calculation relevant to the issue of lon- Y i radii of the corresponding pϩ1d U(N) SYM theory, when- gitudinal momentum transfer physics. ever it is well defined in this limit, i.e., for pр3. As we were finalizing the manuscript, a paper discussing For pϾ3, we see from Eq. ͑6͒ that ¯g ϱ, the dilaton at related issues ͓16͔ came to our attention. s infinity diverges ͑i.e., in the UV of the field→ theory, according to Maldacena’s conjecture ͓8͔͒; this is a statement of the II. A COUPLE OF CONJECTURES non-renormalizability of the corresponding SYM: New phys- A. The matrix conjecture ics sets in the UV. For pϭ4, the D-branes physics is the IR limit of the six-dimensional ͑2,0͒ theory; while for pϭ5, it is A convenient way to summarize the matrix theory conjec- that of a weakly coupled IIB Neveu-Schwarz 5-brane NS5- p ͑ ture is to say that DLCQ M theory on T with N units of brane͓͒18,19͔. We ignore hereafter all cases with pϾ5. In longitudinal momentum is a particular regime of an auxiliary summary, only at low enough energies the 4ϩ1d and 5 ‘‘M¯ theory’’ which freezes the dynamics onto a subsector of ϩ1d SYM yield a proper coarse-grained description of the that theory. Consider such an M¯ theory, with eleven- needed dynamics. ¯ ¯ ¯ dimensional Planck scale l pl ͓which we denote (M, l pl)͔ on a ¯ ¯ B. The Maldacena conjecture pϩ1d dimensional torus of radii Ri , iϭ1...p, and R the ‘‘M theory circle’’ of reduction to type IIA string theory, in It is proposed ͓8͔ that in the limit ͑6͒, one can identify the the limiting regime physics of the SYM QFT at different energy scales with the supergravity solution that is cast by the branes, whenever ¯l 2 ¯l such a solution is well defined. One is to identify string ¯ pl pl l pl 0, with xϵ and y iϵ fixed, ͑3͒ theory excitations of the supergravity background with those → ¯ ¯ R Ri of the quantum field theory ͑QFT͒; this is essentially a correspondence between closed and open string dynamics. and N units of KK momentum along ¯R. It is proposed that Here, we will study finite temperature physics. We will ͓7,17͔ therefore make use of the thermodynamic version of the

044035-3 MIAO LI, EMIL MARTINEC, AND VATCHE SAHAKIAN PHYSICAL REVIEW D 59 044035 above statement, which identifies the finite temperature B. Validity of SYM vacuum of the SYM describing N Dp branes with the geom- Given that we are working with matrix theory on T4 and etry of the horizon region of the near extremal supergravity T5, the first question that must be addressed concerns the solution the branes cast about them, whenever such a solu- validity of the description, given that SYM 4ϩ1d and 5 tion is sensible ͓9͔. In particular, we can extract the thermo- ϩ1d are non-renormalizable. New degrees of freedom are dynamics of SYM at finite temperature in its non- required to make sense of the SYM dynamics as we probe it perturbative regimes. Correlation functions in the SYM in the UV, i.e. as we navigate outward in the corresponding probe different distances from the horizon in the supergrav- supergravity solution. These new degrees of freedom for ity solution as one changes the separation of operator inser- SYM 4ϩ1d and 5ϩ1d are associated with the onset of tions relative to the SYM correlation length ͑thermal wave- strong coupling dynamics; the validity of the theories at dif- length͒; coarse graining the SYM theory to lower energies ferent energy scales is then determined by looking at the size corresponds to moving towards to the center of the super- of the dilaton vev at different locations in the supergravity gravity solution, up until the near extremal horizon ͓20͔. solution. For finite temperatures, physics at the thermal Our strategy will be to use the Maldacena conjecture as a wavelength of the SYM is identified with physics at the ho- tool, to study the thermodynamics of matrix strings and rizon of the near extremal solution ͓9,21͔ black holes; and conversely, to learn about the phase diagram of supersymmetric Yang-Mills theory on the torus. U͑7Ϫp͒/2 2 2 ͑pϪ3͒/2 2 dshorϭ␣Ј d␴i ϩgYͱd pNU d⍀8Ϫp . ͩ g ͱd N ͪ III. A THERMODYNAMIC ROAD MAP Y p ͑12͒ A. Preliminaries We are looking at a fixed time and radial slice; the radial A DLCQ IIA theory descends from the DLCQ M theory variable is U, the ␴ ’s are coordinates along the brane with described above; we choose string scale compactification i identification ␴iϳ␴iϩ⌺i ; d p is a numerical coefficient; and Uo is the location of the horizon, related to the SYM entropy Riϳls for iϭ1...pϪ1, ͑7͒ ͓9͔ with pϪ9 2 Ϫ3 Ϫ2 2 Uo ϳ͑gY ͒ S NV . ͑13͒ 3 3 R pϭgsls l ϭgsl , ͑8͒ pl s The dilaton VEV is and a perturbative IIA regime g2 d N ͑3Ϫp͒/4 ␾ 2Ϫp 2 Y p e ϭ͑2␲͒ gY 7Ϫp . ͑14͒ gsϽ1. ͑9͒ ͩ U ͪ

We can in principle relax Eq. ͑7͒ at the expense of introduc- The finite temperature vacuum of the 4ϩ1d and 5ϩ1d ing new state variables, and a more complicated ͑and richer͒ SYM is a valid thermodynamic description of the DLCQ IIA phase diagram; for simplicity, we will stick to this ‘‘IIA theory ͑by the two conjectures stated earlier͒ when regime.’’ Using the equations in the previous section, we can ␾ ͑8Ϫp͒/͑7Ϫp͒ Ϫ1 write the dictionary between our IIA theory and the matrix e ͉U Ӷ1 SӶN gs . ͑15͒ SYM o ⇒ Note that this is a purely geometric statement, in terms of the 2 pϪ2 pϪ3 gYϭ͑2␲͒ ͑ags͒ , horizon area and string coupling; it will be seen to be insen- sitive to finite size effects due to the transverse torus. We ⌺iϭgsa for iϭ1...pϪ1, then choose to work on a two dimensional cross section in the S-gs plane of the thermodynamic phase diagram, with ⌺ pϭa, ͑10͒ fixed Nӷ1. In principle, one is to take the thermodynamic limit N ϱ with N/S fixed, to see criticality; transition be- pϪ1 pϪ1 p → Vϵ⌺i ⌺pϭgs a , tween phases at finite N discussed here are smooth cross- overs. It is expected that, in the infinite N limit, the physics with tends to the appropriate critical behavior. Let us for a while ignore the effects of the transverse ␣Ј aϵ . ͑11͒ torus. In the regime where the curvature of the supergravity Rϩ solution at the horizon is less than the string scale,

͑pϪ6͒/͑pϪ3͒ Ϫ1 We chose Eq. ͑9͒ so that we have ⌺iϽ⌺ p , simplifying our SӷN gs , ͑16͒ analysis later. We study finite temperature physics of this IIA theory the SYM statistical mechanics obeys the equation of state with the finite temperature vacuum of the corresponding ͓9,21,22͔ SYM. As mentioned in the introduction, we confine our pϪ9 2͑pϪ7͒ 2 pϪ3 7Ϫp 5Ϫp analysis to pϭ4 and pϭ5. Eint ϳS ͑gY͒ N V . ͑17͒

044035-4 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035

FIG. 3. The entropy S versus gs phase diagram showing the FIG. 4. For the convenience of the reader, we reproduce Fig. 1: region of validity of the SYM description, and the boundary be- The proposed thermodynamic phase diagram for the pϩ1d SYM tween the free and interacting phases, ignoring finite size effects. on the torus, i.e., the DLCQ IIA theory. We assume N,Sӷ1, and gsϽ1, and fix N for a given diagram. SϳgϪ2 . ͑19͒ Beyond this regime, we have weakly coupled SYM, roughly s 2 a free gas of N gluons. These statements are graphically Ϫ1/2 This is a statement independent of N and p.AtgcϳN summarized in Fig. 3. Ϫ1/4 ϳNosc , where Nosc is the string oscillator level, there exists Our assumption that finite size effects, due to the compac- an interesting critical point. tification of the ␴i variables, are of no relevance is the state- We next deal with finite size effects in the interacting p ment that, for high enough temperature ͑i.e. for thermal ϩ1d SYM phase which are due to the other radii ⌺i . This is wavelengths short enough with respect to the effective sizes problematic since, unlike the free case, the strongly coupled of the torus͒, the local thermodynamics is similar to that of SYM may acquire at finite temperature a nontrivial vacuum; the uncompactified case. The rest of this section is the study for example, a vacuum characterized by a holonomy sewing of the breakdown of this regime; we would like to paint over branes together. It is in such a regime that Susskind observed Fig. 3 new phases arising due to the compactification of the for the case pϭ3 that the effective size of the SYM box is background. In particular, we will argue that the geometry bigger than ⌺ ͓6͔. In the spirit of Susskind’s transition, and describing the interacting pϩ1d SYM phase gets modified i inspired by the existence of a matrix string phase on the left due to the two transition mechanisms outlined in the Intro- of the phase diagram, we suggest that the finite size effect duction; consequently, the correspondence line of equation can be probed by comparing the Gibbs energies of the inter- ͑16͒ is changed. acting 1ϩ1d and pϩ1d phases. Using Eq. ͑17͒, we get a critical line at C. Finite size effects Ϫ1 A finite size effect in the supergravity regime that was SϳͱNgs , ͑20͒ determined in the work of ͓2,3,4,5͔ is the effect of the DLCQ independent of p and matching2 onto the ‘‘triple point’’ of radius Rϩ on the geometry. We saw from Eq. ͑2͒ that a black hole is localized longitudinally when NϾS. The black hole correspondence. Even so, for these values of S, N and gs , equation of state is the 1ϩ1d SYM is not described by the supergravity solution whose equation of state we used. The analogue of Fig. 3 with 2 pϪ9 pϪ9 N pϭ1 has the correspondence curve on the strong coupling E ϳE . ͑18͒ ␾ bh int ͩ S ͪ side of the line where e ϭ1; in other words, the D string supergravity solution is strongly coupled, as noted in ͓9͔. The process of minimizing the Gibbs energies between the However, the S-dual is the weakly coupled supergravity so- black hole and interacting SYM phases yields a black hole lution of a fundamental IIB string source; its equation of phase as in Fig. 4, independent of p, which is the BFKS state is the same as the one used above, given that the en- observation ͓2,3,4͔; but we now see that the Maldacena con- tropy is to be calculated in the Einstein frame. The curvature jecture justifies the procedure. at the horizon of the S-dual solution becomes of order the The Schwarzschild black hole geometry will become string scale at precisely Eq. ͑20͓͒9͔, beyond which a matrix stringy when its curvature near the horizon becomes of the string description emerges. We can further check the correct- order of the string scale; the emerging state is a matrix string in the matrix conjecture language, i.e. a 1ϩ1 state with ZN 2 holonomy on ⌺ p . Minimizing the Gibbs energy between the Figuratively speaking; we have dropped numerical coefficients in matrix string and matrix black hole phases leads to the this analysis; strictly speaking, this critical ‘‘point’’ may be a mani- Horowitz-Polchinski correspondence curve ͓1,13͔ fold of dimension greater than 0.

044035-5 MIAO LI, EMIL MARTINEC, AND VATCHE SAHAKIAN PHYSICAL REVIEW D 59 044035 ness of this conclusion by matching the pϩ1d interacting sees a transition to the matrix black hole phase as the tem- SYM gas equation of state with that of the matrix string, the perature is lowered only if ͑in the IIA variables͒ latter being the dominant phase on the other side of this correspondence curve. The result is again Eq. ͑20͒. We con- Ϫ1/2 clude that the pϩ1d interacting SYM makes a transition to a gsϾN . ͑23͒ matrix String at Eq. ͑20͒. This is shown in Fig. 4. From the supergravity side, we note that, both the p ϩ1d SYM 1ϩ1d SYM transition and the matrix black This is clear from Fig. 4. → hole matrix string transition are correspondence regimes Finally, we note that we assumed above that there exists a → where the geometry, that of a near extremal fundamental well defined matrix string description for NϽS. In this re- string and that of a black hole respectively, has curvature at gime, the thermal wavelength on the matrix string is smaller the horizon of order of the string scale ͓1͔. On the other than the UV cutoff imposed by the discretized nature of the hand, the BFKS transition is that of longitudinal localization matrices. Our procedure may be equivalent to an analytical of the supergravity solution ͓5͔. continuation of the matrix string phase into a regime where We can now understand the observation of Susskind from the description may not be fully justified; this is in the same the phase diagram of Fig. 4. From the interacting pϩ1d spirit as the extension of the Van der Waals equation of state SYM side, one can consider the effective box sizes ͑i.e. the into the gas-liquid coexistence region, which one uses to critical thermal wavelengths͒ as one approaches the various identify the emergence of the liquid phase ͓23͔. For small transitions. The effective box size is defined by T 1, c⌺effϳ enough coupling gs , we expect the matrix string to evapo- where as usual the temperature is determined from T rate into a perturbative SYM gas, as shown on Fig. 4. Fur- ϳE/S. This yields for the NϳS transition thermore, the regime NϽS is similar to the Hagedorn regime

2 2/͑9Ϫp͒ ͓24,25͔, in that the temperature remains constant as the sys- ⌺effϳ⌺p͑Ngs ͒ , ͑21͒ tem absorbs heat. We speculate that the NϽS regime of the and for the matrix String/pϩ1d SYM transition matrix string near the triple point is characterized by a coex- istent phase of a string with SYM vapor. We defer a detailed analysis of this issue to future work. ⌺effϳ⌺iͱNϭ⌺pgsͱN. ͑22͒ As a unifying probe for all the transitions, we observe that The bound of Susskind †equation ͑3.5͒ of ͓6͔‡ is simply that, the ‘‘mass per unit charge’’ q deﬁned in Eq. ͑1͒ scales on the starting with the pϩ1d SYM phase at high temperature, one various transition curves as

Matrix String-pϩ1d SYM Transition qϪ1ϳg l → eff s Matrix String-Coexistence Phase Transition qϪ1ϳl → s Matrix String-Black Hole Transition qϪ1ϳg2 l → eff s Black Hole-pϩ1d SYM Transition qϪ1ϳg2/͑9Ϫp͒l → eff s

with the effective coupling IV. COMMENTS ABOUT CHARGED PHASES

In this section, we will focus on the triple point, where the g2 ϵg2N. ͑24͒ eff s correspondence and localization effects coincide, and present general considerations of relevance to singly and doubly From the point of view of the DLCQ string theory charac- charged black holes. We will see that, at the critical point, terized by the parameters gs , ls and N, this scaling on the charged black holes are characterized by N/SϾ1. For the transition curves is a non-trivial signature of a unifying singly charged case, by making use of ’t Hooft holonomies framework underlying the physics of criticality of the theory. on the torus, we will sketch a simple dynamical mechanism 2 Note also that the gs N combination is not the ’t Hooft cou- by which the system can cluster its SYM excitations at the pling of the matrix SYM description, Eq. ͑10͒; recall that gs triple point so as to account for the ratio N/S. is a modulus of the torus compactification. Let us begin by rephrasing part of our previous analysis in From the point of view of field theory, the various transi- a slightly different language, using the IMF formalism. Con- tions that we have identified are predictions about the ther- sider a black hole in D dimensions arising from M theory on modynamics of 4ϩ1d, 5ϩ1d SYM on the torus well into T pϩ1, and define Dϩpϭ10. Denote the size of the M theory non-perturbative field theory regimes. circle by R, and suppose the other circles have the charac-

044035-6 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035

Ϫ␣ teristic sizes Ri as in Sec. II. The horizon radius r0 of a black e R hole of mass M is THϳ ϳ 2 , ͑31͒ r0 r0 l9 l3 ͑DϪ3͒/2 DϪ3 pl pl where ␣ is the rapidity of the boost needed to fit the hole in r0 ϭ Mϳ ͑25͒ RR1¯Rp ͩ R ͪ the longitudinal box. The temperature of emitted quanta must be the same as the temperature of the gas on the string; at the correspondence point where the horizon size is the equating the two expressions ͑30͒ and ͑31͒,wefind 2 3 string scale ls ϳlpl/R. To reach the correspondence point for 1/2 2 Nosc ls a given mass black hole, one tunes the string coupling gs 3/2 ϳ 2 . ͑32͒ ϳ(lpl /R) while holding the size of the compactification N r0 torus ͑other than the M theory circle͒ fixed in string units. Now the boost quantum N is the entropy of the black hole, as Therefore let all the Riϭ␭ls ; one finds is the square root of the oscillator number; the left-hand side dϪ1 Ϫ2 is of order unity, and therefore the black hole is of order the Mlsϳ␭ gs . ͑26͒ string size. This is a rephrasing of the observation ͑19͒ of At this point, the string entropy Sec. III, at the triple point. Moving away from this point by boosting to NϾS is a canonical operation; Sϳ⌺T for a string, and N while T 1/N. Note that the number of S ϳͱN ͑27͒ ⌺ϰ ϰ s osc matrix partons in a thermal wavelength of the matrix string at the correspondence point is always N N/S, in agree- is of the same order of magnitude as the black hole entropy clϳ ment with the proposal of ͓14͔. (G is the D-dimensional Newton constant͒ D For a singly-charged hole, the mass, charge and entropy DϪ2 Ϫ1 are given by SBHϳr0 GD . ͑28͒ 1 To work at the triple point, we demand that the string MϳGϪ1rDϪ3 ch2␥ϩ D 0 DϪ3 coupling is tuned so that the horizon size is of order the ͩ ͪ string scale, as dictated by the correspondence principle, and QϳGϪ1rDϪ3sh␥ch␥ that the black hole is placed in a small box and boosted to the D 0 black hole–black string transition ͓2,3,4,5,26,27͔, so that it SϳGϪ1rDϪ2ch␥ ͑33͒ can be described by matrix theory as a D-brane fluid. At the D 0 latter point, the entropy of an uncharged black hole is related in terms of the ‘‘charge rapidity’’ ␥. The Hawking tempera- to the momentum PϭN/R of the boosted hole by SBHϳN ture in the boosted frame is ϭRP. In terms of general relativity, the effect of the boost is to expand the proper size of the box near the black hole so eϪ␣ that it ‘‘just fits inside.’’ For a small box and a large black THϳ ; ͑34͒ r ch␥ hole, the system is highly boosted at the string-hole transi- 0 tion. In the IMF, the weak-coupling string that ͑according to again equating this temperature with the string temperature the correspondence principle͒ approximates the black hole, ͑30͒ and setting the horizon size equal to the string scale has an energy correctly yields 1 2 1/2 N ELCϳ⌺T ϳ 2 Nosc , ͑29͒ SBHϳNoscϳ . ͑35͒ Pls ch␥ where ⌺ is the length of the string, T is its temperature, Nosc This implies that, for the charged case, even at the BFKS is the oscillator excitation number, and PϭN/R is the lon- point, N/SϾ1. This ratio was interpreted in ͓14͔ as the size gitudinal momentum. From this we find the temperature is of clusters of partons making up the black hole phase. There- given by fore, the charged black hole is to be described at the BFKS point as a gas of clusters of size ch␥. 1/2 We can see a mechanism for this dynamics with the fol- RNosc Tϳ . ͑30͒ lowing argument. Consider the matrix string limit of matrix Nl2 s theory on T2. Matrix string transverse excitations are stored in the scalars of the 2ϩ1 SYM, on the diagonal of the ma- On the other hand, one expects the black hole to emit Hawk- 3 trices, with the eigenvalues sewn by boundary conditions ing radiation at temperature involving the shift operator ͓28͔. Off-diagonal constant modes describe the effect of the W bosons stretched between the strands of the string; their one-loop fluctuations give the 3 Ϫ␣ We remind the reader the IMF kinematics ELCϳMe and P effective gravitational interaction of the bits of the matrix ϳMe␣. string. Given a holonomy describing a singly charged matrix

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2 2 string phase, one can ask what is the interaction potential ch ␥wϩch ␥ p NϳS ͑42͒ between two points on the strands, or between two matrix ch␥ ch␥ elements on the D-string. Fixing a winding charge Q, a mag- w p netic ﬁeld must be turned on between the longitudinal boost quantum and the entropy. Note that SϳN1/2 ϩN1/2 ϰch␥ ch␥ , while the Hawking R osc,L osc,R w p B ϭ Q. ͑36͒ temperature must be THϭ2TLTR /(TLϩTR) 12 N Ϫ1 ϰ(ch␥wch␥ p) . A consistent assignment of worldsheet oscillator numbers with respect to these quantities is This induces a holonomy on the 2-torus, with ’t Hooft style boundary conditions ͓29͔. Extending these conditions onto Ϫ1 DϪ2 2 NL,Rϳ„GD r0 ch͑␥wϮ␥ p͒… , ͑43͒ the scalars Xi, we get

i k i Ϫk so that the oscillator entropy agrees with the black hole en- X ϭV X V tropy ͑41͒, and the Hawking temperature determined from 4 i l i Ϫl Eq. ͑40͒ is of the right order. We note that we have again X ͑␴ϩ⌺͒ϭU X ͑␴͒U , ͑37͒ N/SϾ1 at the triple point due to the presence of Kaluza- Klein ͑KK͒ charges. with U ϭ␦ , V ϭdiag͓exp 2␲i(mϪ1)/N ͔, m ij i,jϪ1 ij „ … Thus, in general it is necessary that the matrix black hole ϭ1,...,N; and klϭ␣ЈQ/R ϭl Q. This configuration de- i s consist of coherent clusters of matrix partons even at the scribes a system consisting of l strings, and requires the van- BFKS transition; and that, at the correspondence point, the ishing of some of the off-diagonal elements between the ma- number of partons in a cluster is the same as the number of trix strands. This latter effect reduces the strength of the strands of the matrix string lying within a thermal wave- interaction of the strings when the off-diagonal modes are length. Both of these facts support the analysis of ͓14͔. integrated, by a Q and l dependent factor. Modeling the system through the interaction of the zero modes, and taking into account the multiplicity factor in the interaction result- V. THE INTERACTING MATRIX STRING ing from the boundary conditions ͑37͒, we find an energy for In this section, we come back to the neutral case, away the gas as a function of l: from the triple point, attempting to probe the dynamics in greater detail from the matrix string side. In ͓14͔,itwas N N 2 Ql G v4 2 s D argued that the matrix black hole phase, as a SYM field E͑l͒ϳ v ϪKl 3 7Ϫd , ͑38͒ lR ͩ l ͪ N R r configuration, can be thought of as a gas of S clusters of D0 branes, the zero modes of the SYM, each cluster consisting with K a numerical coefficient independent of N and l. Ap- of N/S partons. The system is self-interacting through the plying the uncertainty principle and the virial theorem ͓5͔ at v4/r7 interaction, or its smeared form on the torus. the correspondence point r0ϳls then yields the scaling This phase of clustered D0 branes may be an effective description, i.e. thermodynamically strongly correlated re- N ϳch␥, ͑39͒ gions of a metastable state; or more optimistically, it might l be a microscopic description associated with formation of bound states like in BCS theory. We will try here to inves- in agreement with Eq. ͑35͒; i.e., it is energetically favorable tigate the matrix string dynamics so as to reveal the signature for the system to settle into a phase of clusters of matrices of of the clusters as we approach the correspondence curve. The size N/Sϳch␥. aim is to identify a possible dynamical mechanism for black Finally, a doubly-charged hole is obtained in the corre- hole formation, and determine the correspondence curve spondence principle from a string carrying both winding and from such a microscopic consideration. momentum in compact directions. The IMF energy and the In Sec. V A, we derive the potential between two points worldsheet temperatures of left- and right-movers are on the matrix string; in view of the matrix conjecture, we can do this by expanding the DBI action of a D-string in the NoscϩNosc 2 2 L R background of a D-string. We then evaluate the expectation ELCϳ⌺͑TLϩTR͒ϳ 2 Pls value of this potential in the free string ensemble at fixed temperature. In Sec. V B, we analyze the characteristic fea- 1/2 RNL,R tures of the potential, particularly noting the bump for p TL,Rϳ 2 . ͑40͒ ϭ4,5 that we alluded to in the Introduction. In Sec. V C, we Nls At the string-hole transition, one has 4These expressions are somewhat different than those in 1 ; the Ϫ1 DϪ3 2 2 ␣ ͓ ͔ PϳGD r0 ͑ch ␥wϩch ␥ p͒e point is that one is free to adjust the smaller of the two temperatures TL,R without appreciably affecting the entropy or the charges. Our Ϫ1 DϪ2 SϳGD r0 ch␥wch␥ p , ͑41͒ choice is compatible with the Bogomol’nyi-Prasad-Sommerfield ͑BPS͒ limit r 0 with ␥ ϳ␥ and the charges fixed, whereas the 0→ w p from which one finds the relation one in ͓1͔ does not give vanishing TR .

044035-8 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035 comment on the dynamics implied by the potential, particu- 2 Ϫ1/2 2 2 1/2 2 ds10ϭh ͑Ϫdt ϩdx ͒ϩh ͑dxជ͒ larly in regard to the phase diagram derived earlier. e␾ϭh1/2 A. The potential C ϭhϪ1 In this section, we derive the potential between two points 01 on the matrix string by expanding the DBI action for a 7Ϫp r0 D-string probe in the background geometry of a D-string. hϭ ͑47͒ We then check the validity of the DBI expansion, and evalu- ͩ r ͪ ate the expectation value of the potential in the thermal en- ͑recall p is the torus dimension͒ and we define Dϵ7Ϫp. semble of highly excited matrix strings. The details of the The string is taken to have no polarizations on the torus, nor finite temperature field theory calculations are collected in any KK charges. Here, we have followed the prescription in Appendix A. ͓35,36͔, where we T-dualized the D-string solution to a D0 A IIA string in matrix string theory is constructed in a brane, lifted to 11 dimensions, compactified on a lightlike sector of field configurations described by diagonal matrices, direction, and T-dualized the solution to the one above. The with a holonomy in ZN ; a nonlocal gauge transformation 7Ϫp 7Ϫp only change is in replacing 1ϩ(r0 /r) (r0 /r) .By converts this into ’t Hooft-like twisted boundary conditions Gauss’ law, → on the transverse excitations of the eigenvalues of the matrices, as described in detail in ͓28,30,31,32͔. The conclusion is g2 D s 9Ϫp that the eigenvalues are sewn together into a long string, and r0 ϭcD 2 ls , a IIA string emerges as an object looking much like a coil or Rϩ ‘‘slinky’’ wrapped on ⌺ . The self-interactions of this string p ͑2␲͒7 are described by integrating out off-diagonal modes between cDϵ D/2 ⌫͑D/2͒, ͑48͒ the well-separated strands. Alternatively, making use of the 2␲ matrix conjecture, this effective action can be obtained from where we have made use of the needed dualities to express supergravity, by expanding the Born-Infeld action of a things in our IIA description. Putting in the background, we D-string in the background of a D-string.5 We will follow have this prescription to calculate the gravitational self-interaction potential between two points on a highly excited matrix 1 ⌺pN string. SϭϪ ͵ hϪ1͑1ϩhKϩh2V͒1/2ϪhϪ1 The Born-Infeld action for N D-strings is given by ͓33͔ ␣Ј 2 ˙ 2 ϪXץ•ϩXץKϵXЈ ϪX ϭ4 1 SϭϪ d2␴eϪ␾ Tr Det1/2͑G ϩB ϩ2␲␣ЈF ͒ 2¯␣ ¯g ͫ͵ ab ab ab Ј s 2 2 2 ϪX͒ …. ͑49͒ץ͑ ϩX͒ץϪX͒ Ϫ͑ץ•ϩXץ͑„Vϵ4 ͑2͒ ϪN͵ CRRͬ, ͑44͒ We note that, as the limit of the action indicates, we have made use of the ZN holonomy that sews the rings of the where we have assumed commuting matrices so that there is slinky together. Expanding the square root yields the Hamil- tonian no ambiguity in matrix orderings in the expansion, and ¯gs is 1 Ϫ the dilaton vev at infinity. We choose ␴ to have radius ⌺ p , ⌺ N 2 7 p p 1 gs ls turn off gauge and NS-NS fluxes, Hϭ X˙ 2ϩX 2 ϩcЈ X 2 X 2 ͒ Ϫץ͑ ͒ ϩץ͕͑ Ј ͒ D 2 7Ϫp ͑ ͵ 2␣Ј Rϩr BabϭFabϭ0, ͑45͒ 2 2 ϪX͖͒. ͑50͒ץ•ϩXץ͔͑ ϪX͒ץϩX͒ ϩ͑ץϪ͓͑ and choose the static gauge Let us check the validity of the DBI expansion we have X0ϭ␴01, performed. We would like to study dynamics of the string squeezed at most up to the string scale, the correspondence point; setting rϳl in h, we get X1ϭ␴11. ͑46͒ s 2 gsls A single D-string background in the string frame is given by hϳ . ͑51͒ R ͓34͔ ͩ ϩ ͪ From elementary string dynamics ͓Eqs. ͑27͒,͑29͔͒, we have

2 5Note that, for D-string strands closer to each other than the RϩS ͗K͘ϳ , ͑52͒ Planck scale, the W bosons cannot be integrated out of the problem; ͩ l N ͪ s the physics is described by the full non-abelian degrees of freedom. We are assuming here that this ‘‘UV’’ physics does not effect the where brackets indicate thermal averaging at ﬁxed entropy analysis done at a larger length scale. S. It can be shown from the results of the next section that

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2 Ϫ1 ͗V͘maxϳ͗K͘ , ͑53͒ SӶNgs . ͑55͒ and that ͗V͘ will have a deﬁnite maximum for all p.We A glance at Fig. 4 reveals that we are well within the region now see from Eqs. ͑49͒, ͑51͒, ͑52͒, and ͑53͒, that our DBI of interest. expansion is a perturbative expansion in The potential in this expression is the interaction energy between a D-string probe and a D-string source. Using the S residual Galilean symmetry in the DLCQ, and assuming ␧ϭgs . ͑54͒ N string thermal wavelengthsϾ⌺ p ͑the ‘‘slinky regime’’͒,we deduce that the potential between two points on the matrix We then need strings denoted by the labels 1 and 2 is

2 5Ϫp 2 2 2 2 ϪXr͖͒ץ. ϩXrץ͔͑ ϪXr͒ץϩXr͒ ϩ͑ץϪXr͒ Ϫ͓͑ץ͑ ϩXr͒ץ͕͑ gs ls V12ϭKD 2 D/2 , ͑56͒ Rϩ ͑Xr ͒

where B. The thermal free string The thermodynamic properties of the matrix string at in- XrϵX2ϪX1 ͑57͒ verse temperature ␤ are determined by the Green’s function of the Laplacian on the worldsheet torus of sides ͑⌺,␤͒, and KD is a horrific numerical coefficient we are not inter- where ⌺ϵ⌺pN. It is known from conformal field theory ested in. ͑CFT͒ on the torus that this is given by ͓38,39͔ Ideally, one should self-consistently determine the shape distribution of the string in the presence of this self- z ␪ ␶ interaction; however, this is rather too complicated to actu- 1 1ͩ ⌺ ͯ ͪ 1 z 2 G12ϭϪ ln ϩ Im , ͑60͒ ally carry out. To first order in small gs , the effect of the 2␲ ͯ 0 ͯ 2␶ ⌺ ␪Ј͑ ͉␶͒ ͩ ͪ 1 2 potential is to weigh different regions of the energy shell in phase space ͓37͔ by a factor derived from its expectation where value in the free string ensemble. We will discuss the dynamics in the presence of the potential in somewhat more ␤ i ␶ϵi ϵi␶ ϭ . ͑61͒ detail below. For now, in light of this weak-coupling ap- ⌺ 2 S proximation scheme, we would like to calculate the expectation value of the potential in a thermodynamic ensemble con- Here ␤ can be obtained from the free string thermodynamics sisting of a highly excited free string with fixed entropy S. of Eq. ͑30͒. All correlators and their derivatives must even- From the matrix string theory point of view, this is essen- tually be evaluated on a time slice corresponding to the real tially a problem in finite temperature field theory, where we axis in the z plane. will deal with a two dimensional Bose gas ͑ignoring super- Divergences will be seen in correlators due to infinite zero symmetry; the fermion contribution is similar͒ on a torus point energies. The conventional approach is to introduce a with sides ⌺ pN and ␤ϭ1/T, ␤ being the period of the Eu- normal ordering scheme giving the vacuum zero expectation clidean time. Using Wick contractions, we can then express value in such situations, i.e. throwing away disconnected the potential in terms of the free Green’s functions; we defer vacuum bubbles. In our case, the string has a classical back- the details to Appendix A. We get ground due to its thermal excitation. To renormalize finite T correlators, we subtract the zero temperature limit from each l5Ϫp KzzK¯z¯z propagator. This corresponds to 2 s 12 12 V12ϭ␣Dgs D/2 , ͑58͒ Rϩ ͑ϪK12͒ ␦ ␦ ͗ f ͑X͒͘ϳ f ln Z ͓J͔ f ln Z ͓J͔ ͩ ␦Jͪ T → ͩ ␦Jͪ T where ␣D is a dimension dependent numerical coefficient, ␦ ␦ ZT͓J͔ Ϫ f ln ZTϭ0͓J͔ϭ f ln . K12ϵK⌬ϵϪ␣Ј͗X1X2͘ϵϪ␣ЈG12 , ͑59͒ ͩ ␦Jͪ ͩ ␦Jͪ ͩ Z ͓J͔ͪ Tϭ0 ͑62͒ is the Green’s function of the two dimensional Laplacian on zz the torus, and K12 is its double derivative with respect to the From the expression for Z͓J͔, we see that this amounts to z complex coordinate of the Riemann surface representing correcting the Green’s functions as the Euclideanized world-sheet. We refer the reader to Ap- pendix A for the derivation of this equation. K K ϪK . ͑63͒ → T Tϭ0 044035-10 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035

FIG. 5. ϪK⌬ as a function of the string separation parameter x; zz we see the change of scaling from x2 to x. FIG. 6. K⌬ as a function of the string separation parameter x; we see the flattening of the correlation at large x. For small x, small This subtraction removes the divergent zero-point fluctua- relative stretching or motion is implied; for larger x, the flattening indicates a constant correlation in the relative stretching of the tions of nearby points on the string, while leaving the effects string. due to thermal fluctuations. Defining ␣ЈSx for ␶2Ӷ1 z K Ӎ ␣Ј ͑70͒ xϵ , ͑64͒ ⌬ S2x2 for 2␲xӶ1, ⌺ ͫ ␲ we then have, subtracting the zero temperature part, The first line is a well-known result of Mitchell and Turok ͓41͔ calculated originally using the microcanonical en- 1 1 2 1/4 1/2 2 semble. It shows random walk scaling R ϳN x . The K ␣Ј ln gg¯ϩ ͑xϪ¯x͒ , ͑65͒ ͱ͗ ͘ osc ⌬→ ͩ 4␲ 8␶ ͪ second line is new and valid for small separations on the 2 string; it is the statement that within the thermal wavelength with ␤ of the the excited string, the string is stretched, scaling as 2 1/2 ͱ͗R ͘ϳNoscx. This is intuitively expected, as regions on the 1Ϫ2qn cos͑2␲x͒ϩq2n string within the typical thermal wavelength will be strongly ln gϭ ͚ ln n 2 . ͑66͒ nϭ1 ͩ ͑1Ϫq ͒ ͪ correlated in the thermodynamic sense. This change in the scaling is crucial to what we will soon see in the behavior of the potential between strands. ϪK is plotted in Fig. 5. We can now make use of ␶2Ӷ1 for Sӷ1, to write this sum ⌬ as an integral Next, consider the derivatives of the correlators, evaluated on the real axis. We have 2 1 eϪ2␲␶2 dv 1Ϫ2v cos͑2␲x͒ϩv Ϫ2␲␶ Ϫ2␲ix ln gϭ ln 2 . Ϫi␣Ј 1Ϫe 2 2␲␶ ͵0 v ͩ ͑1Ϫv͒ ͪ K ϭ ln . ͑71͒¯ץK ϭ ץ 2 x ⌬ x ⌬ 2 Ϫ2␲␶2ϩ2␲ix ͑67͒ 2͑2␲͒ ␶2 ͩ 1Ϫe ͪ

This integral can be evaluated to yield We also have

1 Ϫ2␲␶2 Ϫ2␲␶2ϩ2␲xi ln gϭ ͑2Li2e ϪLi2e 2␲␶2

Ϫ2␲␶2Ϫ2␲xi ϪLi2e ͒, ͑68͒ where Li2 is the PolyLog function of base 2, related to the Lerch ⌽ function ͓40͔. We then have

␣Ј K ϭ ln g, ͑69͒ ⌬ 2␲ with x here being real, and representing the equal-time separation between two points on the string, xϭx1Ϫx2 ,asa FIG. 7. The potential as a function of x for dimensions pϭ3 and fraction of the total length ⌺ (0ϽxϽ1). The asymptotics are pϭ2.

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Ϫ␣Ј or xK⌬ϭ , ͑72͒¯ץxץ 4␶2 or ␣Ј 1Ϫe2␲␶2 cos͑2␲x͒ ⌺2Kzzϭ⌺2K¯z¯z Ϫ ⌬ ⌬ 4␲␶2 2␲␶2 zz¯ → ␶2 e Ϫ2e cos͑2␲x͒ϩ1 K⌬ 0, ͑73͒ → ␣Ј 1 since we subtract the zero temperature result. The most rel- Ϫ ͑75͒ ␶ e2␲␶2Ϫ1 evant term is 2

2␲␶2 2 2 ␣Ј 1Ϫe cos͑2␲x͒ ␣Ј .K ϭϪ ϩ , ͑again we subtract the zero temperature part͒¯ץK ϭ ץ x ⌬ x ⌬ ␶ e4␲␶2Ϫ2e2␲␶2 cos͑2␲x͒ϩ1 4␶ 2 2 This yields the asymptotics ͑74͒

2 ␲ 2 2 2 ␣ЈS ϩ 5ϩcos͑2␲x͒ csc͑␲x͒ ϩO͑␶ ͒ ␣ЈS for ␶2Ӷ1, 2 zz 12 „ …„ … 2 ͑N⌺ p͒ K⌬ Ӎ → ͑76͒ ͫ␣ЈS4x2 for 2␲xӶ1.

X ) decrease as the separation along the string decreasesץzz Ϫ K⌬ is plotted in Fig. 6 as a function of x. 2 ͑inside a thermal wavelength͒. We observe that: C. The bump potential The bump occurs at separations of 1/S of a fraction of the We now put together Eqs. ͑69͒ and ͑75͒ in the potential of whole length of the string; in the matrix language, this cor- Eq. ͑58͒ to get the asymptotics responds to a bump about matrices of size N/S. The presence or absence of the bump as a function of the 3 ͑8ϪD͒/2 ϪD/2 Rϩ S x for ␶2Ӷ1, number of non-compact space dimensions correlates with the V Ӎg2 ͑77͒ 12 s ␣Ј3N4 ͫ S8ϪDx4ϪD for 2␲xӶ1. observations of ͓13͔, given that in the DLCQ, the light-like direction reduces the number of non-compact dimensions by ϪD/2 one. For pϾ3, V12 0asx 0; at larger x, it decays as x . At the thermal→ wavelength→ xϳ1/S, both expressions give As described in ͓14͔, a matrix black hole can be described by SYM excitations clustered within matrices of size N/S, the location of the bump. Furthermore, we will shortly repro- R3 2 ϩ duce, from scaling arguments regarding the dynamics of this V Ӎg S4. ͑78͒ max s ␣Ј3N4 potential, the two correspondence lines determined from thermodynamic considerations above. Note that in this expression the dimension dependence in the We then conclude that we have identified the characteristic 4 signature of black hole formation in the matrix SYM. power of S conspires to vanish. For pϭ3, V12ϳS for x 0, while for pϽ3, V12 ϱ for x 0; in both of these →latter cases, the potential decays→ as xϪ→D/2 for larger x. D. Dynamical issues and criticality The conclusion can be summarized as follows. For pϭ4 The dynamics of this potential near a phase transition and pϭ5, there exists a bump in the potential of height pro- point is certainly complicated. Intuitively, we expect that as portional to S4 at the thermal wavelength on the string; for we approach a critical point, instabilities develop, an order pϭ3, the bump smoothes to a flat configuration where the parameter fluctuates violently, perhaps related to some mea- difference between the potential at the thermal wavelength sure of the ZN symmetry; it is reasonable to expect the char- separation and at xϭ0 is of order unity. Finally, for pϽ3, acteristic feature of the potential, the confining bump, plays a the bump disappears altogether and the potential blows up at crucial role in the dynamics of the emerging phase. Deferring the origin signaling the breakdown of the description. This a more detailed analysis of these issues to the future, let us potential is plotted in various cases in Fig. 2 of the Introduc- try to extract from these results the scaling of the correspon- tion and Fig. 7. dence curves. The presence or absence of the bump is a result of two First let us motivate the use of the expectation value of competing effects: First of all, the increasingly singular the potential in the free string ensemble. We indicated earlier short-distance behavior of the Coulomb potential ͑56͒ with that this quantity is qualitatively related to the effect of the increasing dimension D; and secondly, the strong correlation interactions, assuming they are weak enough, on the energy X1 shell in phase space covered by the free string. The partitionץ) of neighboring points on the string, which makes

044035-12 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035 function becomes, schematically corrected by a factor in order to reproduce the black hole equation of state; the origin of this correction was argued to H ϩV ͗V͘ H ZϳTr e 0 ϳe 0 Tr e 0, ͑79͒ be interaction processes between the clusters involving the exchange of longitudinal momentum. Under the assumption so that phase space is weighed by an additional factor related that these effects are of the same order as zero momentum to the expectation value of the potential in the free ensemble transfer processes, a correction factor of N/S was applied. ͗V͘0 . This is also similar to the RG procedure applied to the 2D Ising model, where the context and interpretation is Using a chain of dualities, we can quantify the effect of slightly different ͓37͔. longitudinal momentum transfer physics by studying the Using Eq. ͑58͒, the potential energy content of the matrix scattering amplitude in IIB string theory with winding num- string is given by ber exchange. We do this in Appendix B, where we find that, for exchanges of windings up to order N/S, the winding 3 exchange generates an interaction identical to that of zero N⌺p R 4 2 ϩ longitudinal momentum exchange; for higher winding ex- Vϳ d␴Ϫ S gs 3 4 ͑N⌺ p͒v12 , ͑80͒ ͵0 ͩ ␣Ј N ͪ changes, the interactions are much weaker. These winding modes, represent the sections of the matrix string within the where we have integrated over one of the two integrals of the thermal wavelength, N/S worth of D-string windings. Thus translationally invariant two-body potential, and scaled v12 we modify the v12 potential above by the factor N/S, which such that its maximum is of order 1, independent of any state accounts in the scaling analysis for the effect of longitudinal variables; however, the shape of v12 still depends on N and momentum transfer physics in the matrix string self- S. This expression represents the interaction energy between interaction potential. Applying the virial theorem between two points on the coiled matrix string at fixed separation Eq. ͑82͒ and N/S times Eq. ͑81͒ yields the matrix string– ␴Ϫ . From the point of view of matrix theory physics, the matrix black hole correspondence point at string’s fundamental dynamical degrees of freedom are the windings on the coil; we expect a transition in the dynamics of the object when there is a competition between forces on Ϫ2 Sϳgs , ͑84͒ an individual winding. In the present case, the two forces are nearest neighbor elastic interaction and the gravitational interaction. A single string winding being wrapped on ⌺ p as needed. worth of world-sheet, the maximum potential energy it feels We can now interpret our results as follows. The bump can be read from Eq. ͑80͒ potential accounts for the matching of the string phase onto 3 both NϽS and NϾS phases, one involving partons interact- Rϩ v ϳS4g2 N⌺2 , ͑81͒ ing without longitudinal momentum exchange ͓the matrix max s ␣Ј3N4 p string–(pϩ1d) SYM curve in Fig. 4͔, and the other being the matrix black hole phase of parton clusters of size N/S and is due to its interaction with strands a thermal wave- Ͼ1 interacting in addition by exchange of longitudinal mo- length away. Its thermal energy caused by nearest neighbor mentum ͑the matrix string–matrix black hole correspon- interactions is read off Eq. ͑52͒ dence curve of Fig. 4͒. In the latter case, the location of the confining bump correlates with matrices of size N/S. In the K ͗ ͘ former case, the correlations are finer than the UV matrix ␬ϳ ⌺ p . ͑82͒ ␣Ј cutoff; a better understanding of this latter issue obviously needs a more quantitative analysis of the NϽS matrix string The two forces compete when regime. This analysis further substantiates the identification Ϫ1 of the bump potential as the signature of black hole forma- SϳͱNgs . ͑83͒ tion from matrix SYM, as well as justifying the new matrix string-p brane transition microscopically. At stronger coupling, the forces due to the gravitational interaction dominate those of the nearest neighbor stretching and decohere neighboring strands’ velocities. The free string evaluation of the interaction, Eq. ͑58͒, is no longer valid; one ACKNOWLEDGMENTS expects a phase transition to occur. Equation ͑83͒ is our We are grateful to H. Awata for discussions. V.S. is very matching result of Eq. ͑20͒ between the string and pϩ1d grateful to S. Coppersmith for particularly helpful sugges- interacting SYM phase. Here, we are assuming an analytical tions regarding the condensed matter literature. This work continuation of the matrix string phase to the region NϽS in was supported by DOE grant DE-FG02-90ER-40560 and the phase diagram; our suggestion that this region is associ- NSF grant PHY 91-23780. ated with a coexistence phase is consistent with this procedure. To account for the correspondence curve for NϾS,we APPENDIX A: CALCULATION OF THE POTENTIAL now recall that in the discussion of clustered D0 branes of ͓14͔, the virial treatment of the v4/r7 interaction had to be We need to evaluate

044035-13 MIAO LI, EMIL MARTINEC, AND VATCHE SAHAKIAN PHYSICAL REVIEW D 59 044035

2 2 2 2 ϪXr͖͒ץ• ϩXrץ͔͑ ϪXr͒ץϩXr͒ ϩ͑ץϪXr͒ Ϫ͓͑ץ͑ ϩXr͒ץ͕͑ ϵ 2 D/2 , ͑A1͒ V ͳ ͑Xr ͒ ʹ

in the ﬁnite temperature vacuum of the SYM. Let subscripts where the x subscript is integrated over and it is implied to ͑123456͒ denote the argument of X, e.g. X1ϵX(␴1). Writ- be the argument of the J as well. Denoting the number of ing XrϵX5ϪX6 , we will encounter in the numerator only polarizations in the Lorentz indices by d, we get factors of the form Xi Xi X j X j Ϫd/2 1 2 3 4 ␲ d Ϫp2 Ϫ2sp2f2 X j , ͑A2͒ 2 D/2 ϭ ds d pe e ץ X j ץ Xi ץ Xi ץ ␣ 1 ␣ 2 ␤ 3 ␥ 4 ͳ „͑X5ϪX6͒ … ʹ ⌫͑D/2͒ ͵ ͵ with the target indices i, j summed over; ␣, ␤, ␥ are world- 1 ϫ T s͑D/2͒Ϫ1Ϫ p2sD/2T sheet indices Ϯ; and the labels ͑1234͒ are set equal to 5 and ͫ 1 2 2 6 in various ways. By expanding the numerator of Eq. ͑A1͒, we get 3ϫ16 terms of the form claimed. We can write our 2 2 ͑D/2͒ϩ1 desired ‘‘monomial’’ ͑A2͒ as ϩ͑p ͒ s T4ͬ ͑A11͒ i i j j X1X2X3X4 where we have deﬁned A3͒͑ . 4ץ 3ץ 2ץ 1ץ ␣ ␣ ␤ ␥ ͑X ϪX ͒2 D/2 ͳ „ 5 6 … ʹ d2 d d T ϵ K K ϩ K K ϩ K K ͑A12͒ Consider 1 4 12 34 4 13 24 4 23 14

i i j j Ϫd/2 X X X X ␲ ϱ T2ϵdK1K2K34ϩdK3K4K12ϩK1K3K24ϩK1K4K23 1 2 3 4 d ͑D/2͒Ϫ1 2 D/2 ϭ ds d ps ͳ ͑X5ϪX6͒ ʹ ⌫͑D/2͒ ͵0 ͵ „ … ϩK2K3K14ϩK2K4K13 ͑A13͒ 2 ˜ Ϫp i i j j ͐J•X ϫe ␦1␦2␦3␦4͗e ͘ T4ϵK1K2K3K4 . ͑A14͒ Ϫd/2 ␲ ϱ Evaluating the s integral, we get ϭ ds ddps͑D/2͒Ϫ1 ⌫͑D/2͒ ͵0 ͵ i i j j Ϫd/2 X1X2X3X4 ␲ 2 2 d Ϫp 2 ϪD/2 Ϫp i i j j ⌬ 2 D/2 ϭ D/2 2 D/2 d pe ͑p ͒ ϫe ␦1␦2␦3␦4e ͑A4͒ ͳ „͑X5ϪX6͒ … ʹ 2 ͑ f ͒ ͵ where D ͑Dϩ2͒D ϫ T Ϫ T ϩ T . ͫ 1 8 f 2 2 16͑ f 2͒2 4ͬ ˜JiϵJiϩ2iͱs ␦͑␴Ϫ␴ ͒Ϫ␦͑␴Ϫ␴ ͒ pi ͑A5͒ „ 5 6 … ͑A15͒ and Evaluating the p integrals, we get

1 1 i i j j Ϫd/2 ˜ ˜ 2 2 X1X2X3X4 ␲ dϪD ⌬ϵ JKJϭ JKJϩiͱs J pKxϪ2sp f . 4 ͵ 4 ͵ ͵ • 2 D/2 ϭ ⍀dϪ1⌫ ͑X ϪX ͒ 2͑D/2͒ϩ1͑ f 2͒D/2 ͩ 2 ͪ ͳ 5 6 ʹ ͑A6͒ „ … D ͑Dϩ2͒D We have deﬁned ϫ T Ϫ T ϩ T ͫ 1 8 f 2 2 16͑ f 2͒2 4ͬ KxϵKx5ϪKx6 ͑A7͒ ͑A16͒

2 f ϵKϪK56 . ͑A8͒ where ⍀dϪ1 is the volume of the dϪ1 unit sphere. Going back to Eq. ͑A3͒, we need to differentiate T1 , T2 Here Kab means K(aϪb), the Green’s function of the two and T4 , according to the map 1234 ␣␣␤␥. Let us denote dimensional Laplacian the derivatives by superscripts on the→K’s. We then have

2 KabϵϪ␣Ј͗XaXb͘, ͑A9͒ d d d T␣␣␤␥ϭ K␣␣K␤␥ϩ K␣␤K␣␥ϩ K␣␤K␣␥ , 1 4 12 34 4 13 24 4 23 14 and KϵKaa . The rest is an exercise in combinatorics, mak- ͑A17͒ ing use of T␣␣␤␥ϭdK␣K␣K␤␥ϩdK␤K␥K␣␣ϩK␣K␤K␣␥ϩK␣K␥K␣␤ 1 2 1 2 34 3 4 12 1 3 24 1 4 23 i ⌬ i i ⌬ ␦ e ϭ GaxJ ϩiͱsp Ka e ͑A10͒ ␣ ␤ ␣␥ ␣ ␥ ␣␤ a ͫ2 ͵ ͬ ϩK2K3K14 ϩK2K4K13 , ͑A18͒

044035-14 BLACK HOLES AND THE SYM PHASE DIAGRAM PHYSICAL REVIEW D 59 044035

␣␣␤␥ ␣ ␣ ␤ ␥ each term in Eqs. ͑A17͒–͑A19͒, we have 16 terms associated T4 ϭK1K2K3K4 . ͑A19͒ with taking a map from ͑1234͒ to a sequence of 5’s and 6’s. We have used here the translational invariance and evenness This combinatorics yields of the Green’s function to interpret the derivatives as differ- entiations with respect to the argument iϪ j of the Green’s d functions ͑and therefore note some ﬂip of signs͒; further- K␣␤K␣␥ϩ K␣␣K␤␥ X j 56 56 2 56 56 ץ X j ץ Xi ץ Xi ץ more, we assume that K, K␣ and K␣␤ are zero, i.e. because ␣ r ␣ r ␤ r ␥ r 2 D/2 Ӎ D/2 . of subtraction of the zero temperature limits, or throwing ͳ ͑Xr ͒ ʹ ͑ϪK56͒ away bubble diagrams. This, it turns out, is not necessary for ͑A20͒ the potential we calculate, since all expressions would have ␣␤ ␣␤ come out as differences, say K12 ϪK ; it is just convenient Note that T2 and T4 cancelled; we have also dropped nu- for notational purposes to throw them out from the start. We merical coefﬁcients. There are three terms in Eq. ͑A1͒ of this ␣␤ ␣␤ ␣ ␣ ␣ also note the identities K12 ϭK21 and K5 ϭϪK56ϭK6 . For type; this yields

d d KϩϪKϩϪϩ KϩϩKϪϪϪ ϩ1 ͑KϩϩKϩϪϩKϪϪKϩϪ͒ 56 56 2 56 56 ͩ2 ͪ 56 56 56 56 Ӎ D/2 . ͑A21͒ V ͑ϪK56͒

Using the equation of motion ͑delta singularity subtracted͒ 1 ϩϪ K␣␤␥␦ϭϪ ͑ST␩␣␥␩␤␦ϩSU␩␤␥␩␣␦ϩTU␩␣␤␩␥␦͒ K56 ϭ0, we get 2 KϩϩKϪϪ ␣ ␥ ␤␦ ␤ ␦ ␣␥ ␣ ␦ ␤␥ ␤ ␥ ␣␦ 56 56 ϩS͑k4k2␩ ϩk3k1␩ ϩk3k2␩ ϩk4k1␩ ͒ Ӎ D/2 . ͑A22͒ V ͑ϪK56͒ ␥ ␣ ␤␦ ␦ ␤ ␣␥ ␤ ␣ ␥␦ ␥ ␦ ␣␤ ϩT͑k4k2␩ ϩk3k1␩ ϩk4k3␩ ϩk1k2␩ ͒ Using Euclidean time i␶ϭt, we have ␴Ϯϭ␴Ϯtϭz,¯z;ﬁ- ϩU͑k␣k␦␩␤␥ϩk␥k␤␩␣␦ϩk␣k␤␩␥␦ϩk␥k␦␩␣␤͒. nally, we get for Eq. ͑56͒ 2 3 4 1 4 3 2 1 ͑B2͒ l5Ϫp KzzK¯z¯z 2 s 12 12 V12ϭ␣Dgs D/2 . ͑A23͒ This gives the amplitude Rϩ ͑ϪK12͒ 4 4 4 Amϳ͓͑SϩT͒ ϩS ϩT ͔ APPENDIX B: LONGITUDINAL MOMENTUM ⌫͑ϪS␣Ј/4͒⌫͑ϪT␣Ј/4͒⌫͑ϪU␣Ј/4͒ TRANSFER EFFECTS ϫ . ⌫͑1ϩS␣Ј/4͒⌫͑1ϩT␣Ј/4͒⌫͑1ϩU␣Ј/4͒ Consider the scattering of two wound strings in IIB theory with winding number exchange. We will ﬁnd that, in the ͑B3͒ regime of small momentum transfer, the interaction is Cou- We want to accord winding n1 , n2 , n3 and n4 to the four lombic for resonances involving low enough winding num- strings, on a circle of radius R; without any momenta along ber exchange, and much weaker otherwise; furthermore, the this cycle, we can extract easily this process from the ampli- Coulombic interaction is winding number independent, and tude above by the cumulative strength of this potential suggests modifying the matrix string potential by a factor of N/S for NϾS. sϭSϩM 2, ͑B4͒ For simplicity, consider the polarizations of the external states to be that of the dilaton, and T-dualize the momentum tϭTϩm2ϵϪq2, ͑B5͒ in the compact direction to winding number. The resulting four string amplitude is given by ͓42͔ with

␣␤␥␦ 2 A ϳK K R͑n1ϩn2͒ m ␣␤␥␦ M 2ϵ , ͑B6͒ ␣Ј ⌫͑ϪS␣Ј/4͒⌫͑ϪT␣Ј/4͒⌫͑ϪU␣Ј/4͒ ͩ ͪ ϫ , ⌫ 1ϩS␣ /4 ⌫ 1ϩT␣ /4 ⌫ 1ϩU␣ /4 2 ͑ Ј ͒ ͑ Ј ͒ ͑ Ј ͒ R͑n3Ϫn2͒ m2ϵ . ͑B7͒ ͑B1͒ ͩ ␣Ј ͪ

2 2 2 2 where SϵϪ(k1ϩk2) , TϵϪ(k2ϩk3) , UϵϪ(k1ϩk3) , For large m1 ,m2 , and small m, q is the spatial momentum with SϩTϩUϭ0, and transfer between the strings in the center-of-mass frame.

044035-15 MIAO LI, EMIL MARTINEC, AND VATCHE SAHAKIAN PHYSICAL REVIEW D 59 044035

2 d/2Ϫ1 Thus Mӷm, and we are in the non-relativistic regime Ecm m ␲ 2 V͑2͒ϳT͑2␲͒d/2 eϪmr, ͑B15͒ ӷq . From Eqs. ͑B4͒ and ͑B5͒, we see that SӷT. Using this eff ͩ r ͪ ͱ2mr and the identities ⌫(z)⌫(1Ϫz)sin(␲z)ϭ␲ and ⌫(1ϩz) ϭz⌫(z), one obtains the amplitude (1) which is weaker than Veff since we have mrӷ1. Finally, we 2 2 2 2 2 2 2 have Amϳ͑sϪM ͒ sin„␲͑q ϩm ͒␣Ј/4…„⌫͓͑q ϩm ͒␣Ј/4͔… . ͑B8͒ sin m2 1 V͑3͒ϳT . ͑B16͒ In the energetic regime considered, eff m4 rd

2 2 sϪM ϳm1m2vrelϵͱ , ͑B9͒ In addition to a larger power in r, we have mӷ1; this inter- T action is much weaker than Eqs. ͑B14͒,͑B15͒, especially af- where vrel is the relative velocity of strings 1 and 2 in the lab ter averaging over a range of winding transfers m. frame. We conclude that, for mrϭRr(n3Ϫn2)/␣ЈӶ1, we have a Equation ͑B8͒ has poles at q2ϩm2ϭ4n/␣Ј with nр0. Coulombic potential independent of the winding exchange We consider scattering processes probing distances r much m; for mrӷ1, we have much weaker potentials. This implies larger than the string scale, qmaxϳ1/rӶ1/ls ; we also assume that in a gas of winding strings bound in a ball of size at that it is possible to have RӶls , which we will see is nec- most of order the string scale, the dominant potential is Cou- essary. Given that these poles space the masses of the reso- lombic with a multiplicative factor given by w0ϵ␣Ј/(Rr), nances by the string scale, the dominant term to the ampli- provided a mechanism restricts winding exchange processes tude is the one corresponding to the exchange of a wound to n3Ϫn2Ӷn1ϩn2 . ground state, i.e. the nϭ0 pole. Measuring quantities in The S-dual of this amplitude describes the scattering of string units, the amplitude then becomes wound D-strings at strong coupling, with winding number exchange. Under a further T duality, and lifting to M theory, sin͑q2ϩm2͒ A ϳ . ͑B10͒ this amplitude encodes a good measure of the effects of lon- m T ͑q2ϩm2͒2 gitudinal momentum exchange in the problem of a self- interacting matrix string. The bound on the winding number The effective potential between the strings is the Fourier translates in our language to transform of this expression with respect to q. Let us consider various limits. Take mӶq; we then have mӶ1. The ¯␣Ј¯gs R11 N w ϭ ϭ ϳ , ͑B17͒ amplitude becomes 0 ⌺r r S

A͑1͒ϳ T . ͑B11͒ i.e. the resolution in the longitudinal direction. We also note m q2 that, under this chain of dualities, the string scale used to set a bound on the impact parameter r transforms as ␣Ј ␣Ј, Next consider mӷq, but mӶ1. The amplitude becomes where the latter string scale is that of the matrix string.→ This justiﬁes our implied equivalence between the scale of r and ͑2͒ Am ϳ 2 T 2 . ͑B12͒ that of the size of the black hole. q ϩm In the single matrix string case we study, we saw that Finally, for mӷq and mӷ1, we have a constant regions of size N/S were strongly correlated and ‘‘rigid’’ in a statistical sense. The self-interaction of the large string will sin m2 then involve processes of coherent exchange of D-string A͑3͒ϳ . ͑B13͒ winding up to the winding number N/SӶN. For larger wind- m T m4 ing, the D-string is not coherent; one expects a suppression The effective potentials are then (dϵ9Ϫp) both from the emission vertex and from the highly off-shell propagator. We saw above that all such processes, up to T N/S, are of equal strength and scale Coulombically. This V͑1͒ϳ ddqeiq•xA͑1͒ϳ . ͑B14͒ eff ͵ m rdϪ2 implies that the potential between the string strands calculated from the DBI expansion must be enhanced by a factor The result is a Coulomb potential, independent of m. The of N/S for NϾS, and justiﬁes the scaling arguments used in second case gives Sec. V D.

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