fulltext

Heterotic-string thermofield dynamics (*)

H. FUJISAKI(1), K. NAKAGAWA(2)(**) and S. SANO(1)(***) (1_{) Department of Physics, Rikkyo University - Tokyo 171, Japan}

(2_{) Faculty of Pharmaceutical Sciences, Hoshi University - Tokyo 142, Japan} (ricevuto il 2 Dicembre 1996; approvato il 3 Aprile 1997)

Summary. — Physical aspects of the thermofield dynamics of the D 410 heterotic

thermal string theory are exemplified through the infrared behaviour of the one-loop dual symmetric cosmological constant in association with the global phase structure of the thermal string ensemble.

PACS 11.10.Gh – Renormalization.

PACS 12.10 – Unified field theories and models.

Elaboration of thermal string theories based upon the thermofield dynamics (TFD) [1] has gradually turned out to be a good practical subject of high-energy physics [2-9]. In a previous paper of ourselves [8], we have succeeded in shedding some light upon the global phase structure of the thermal string excitation in proper reference to the thermal duality relation [10-13] for the D 426 closed bosonic thermal string theory within the TFD framework. In the present communication, physical aspects of the D 410 heterotic thermal string theory based upon the TFD algorithm are exemplified through the infrared behaviour of the one-loop cosmological constant in respect of the thermal duality symmetry. The global phase structure of the TFD thermal string amplitude is then examined à la O’Brien and Tan [10] on the basis of the thermal stability of modular invariance.

Let us start with describing the one-loop cosmological constant L(b) at any finite temperature b21 4 kT as L(b) 4 a 8 2 mlim2 K 0 Tr

y



Q m2 dm2

₍

DbB(p , P ; m2) 1DbF(p , P ; m2)

)

z

(1)

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail address: nakagawaHhoshi.ac.jp

(***) E-mail address: sanoHrikkyo.ac.jp

for the D 410 heterotic thermal string theory in the TFD framework, where the string tension s is expressed in terms of the slope parameter a 8 as s41/2pa8, pm_{reads loop}

momentum and PI

lie on the even self-dual root lattice L 4G83 G8for the exceptional group G 4E83 E8[14], respectively. Here the thermal propagator D

b

B[ F ](p , P ; m2) of the free closed bosonic [fermionic] string is expressed at D 410 as

(2) DbB[ F ](p , P ; m2) 4



2p p df 4 pe if(N 2a2N–1 a–21 / 2 Q

!

I 41 16 (PI)2) 3 3

u

y

_[2]1



0 1 dx 1 1 2 n 40

!

Q d [a 8/2Qp21 a 8 / 2 Q m21 2(n 2 a) ] ebNp0N 2 [1]1



c dx

z

₃ 3xa 8 / 2Qp21 N 2 a 1 N – 2 a–1 1 / 2 Q

!

I 41 16 (PI)21 a 8 / 2 Q m22 1

v

_, where N [N–] denotes the number operator of the right- [left-] moving mode, the intercept parameter a [a–_{] of the right- [left-] mover is eventually fixed at a 40 [a}–_{4 1 ]} and the contour c is taken as the unit circle around the origin. The D 410 thermal cosmological constant L(b) is immediately reduced to

(3) L_{(b) 42} ia 8 4 mlim2_{K 0}



Q m2 dm2



2Q Q dD p ( 2 p)D

!

n 40 Q d

k

a 8 2 p 2 1 a 8 2 m 2 1 2(n 2 a)

l

3 3

g

1 ebNp0N 2 1 1 1 ebNp0N 1 1

h

P

!

I_L



2p p df 4 pe 2if

g

a 2a–11 / 2 Q

!

I 41 16 (PI₎2

h

3 3



c dx xa 8 / 2Qp22 a 2 a–11 / 2 Q

!

I 41 16 (PI₎2 1 a 8 / 2 Q m22 1_{tr [e}if(N 2N–) xN 1N–] . Let us turn our attention to explicit calculation of the D 410 thermal amplitude L(b). The integration over the time-like component p0in eq. (3) yields

(4) L_{(b) 42} i 4 mlim2_{K 0}



Q m2 dm2



2Q Q dD 21p ( 2 p)D

!

n 40 Q 1 vn(m2) 3 3

u

1 ebvn(m2) 2 1 1 1 ebvn(m2) 1 1

v

P

!

I_L



2p p df 4 pe 2if

g

a 2a–11 / 2 Q

!

I 41 16 (PI₎2

h

3 3



c dx xa 2a–11 / 2 Q

!

I 41 16 (PI₎2_{2 2 n 2 1} tr [eif(N 2N–)_xN 1N–_{] ,} where vn(m2) 4

k

p21 4(n 2 a) /a 8 1 m2 . (5)

The integration over m2_{brings forth} (6) L(b) 4 i b



2Q Q dD 21_p ( 2 p)D

!

n 40 Q

!

l 40 Q ₁ l [ 1 2 (21) l_{] e}2bvn( 0 ) l 3 3

!

PI_L



2p p df 4 pe 2if

g

a 2a–11 / 2 Q

!

I 41 16 (PI₎2

h



c dx xa 2a–11 / 2 Q

!

I 41 16 (PI₎2 2 2 n 2 1_{tr [e}if(N 2N–) xN 1N – ] at the limit m2

K 0, with the aid of an expansion technique of the Bose-Einstein and Fermi-Dirac distribution factors. Equation (6) is then reduced to

(7) L_{(b) 4} i 4 p( 4 p 2 a 8)2D/ 2



2p p df 4 pe 2if(a 2 a–)



0 Q dt2 tD/ 2 112 e4 pat2 3 3



c dx xa 2a–21

!

n 40 Q exp [22n(2pt21 ln x) ]

!

PIL (xe2if₎1 / 2 Q

!

I 41 16 (PI₎2 3 3tr [eif(N 2N – )_xN 1N–_]

!

l  Z ; odd exp

k

₂ b 2 4 pa 8 t2 l2

l

_, where use has been made of the formula [2]



0 Q dp p2 n e 2bkp21 m2

k

p2_{1 m}2 4 G (n 11O2) 2 kp g n



0 Q dt2 tn 112 e2m2t2/g 2b2g/4 t2 (8)

with g 42s41/pa8. The summation over n in eq. (7) yields (9) L_{(b) 4} i 8 p( 4 p 2 a 8)2D/ 2



2p p df 4 pe 2if(a 2 a–)



0 Q dt2 tD/ 2 112 e2 pt2( 2 a 11)



_{dx x}a 2a–11₃ 3

g

1 x 2e22 pt2 2 1 x 1e22 pt2

h

!

PIL (xe2if₎1 / 2 Q

!

I 41 16 (PI)2 3 3tr [eif(N 2N – )_xN 1N–_]

!

l  Z ; odd exp

k

₂ b 2 4 pa 8 t2 l2

l

_. We are then eventually led to the modular parameter integral representation of L(b) at

D 410 as follows [10]: L_{(b) 42 8(2pa8)}2D/ 2



E d2 t 2 pt22 Kh(t–, t ; D)

!

l  Z ; odd exp

k

₂ b 2 4 pa 8 t2 l2

l

_, (10)

where (11) Kh(t–, t ; D) 4 (2pt2)2(D 2 2 ) / 2e2 pi t –

k

1 1480

!

m 41 Q s7(m) z–m

l

3 3

»

n 41 Q ( 1 2z–n₎2D 2 14

g

1 1z n 1 2zn

h

D 22 , t [2] 4t1 1 [2]it2, z 4xeif4 e2 pit, z –

4xe2if4e22 pi t–, a [ a–_{] has been fixed at a40 [a}–_{41 ],}

E means the half-strip integration region in the complex t-plane, i.e. 21/2 Gt1G 1 / 2;

t2D 0, and full use has been made of an explicit expression of the theta-function UG83 G8 of the root lattice G83 G8[14]. Accordingly, the D 410 thermal amplitude bL(b) is identical in every detail with the “E-type” representation of the thermo-partition function Vh(b) of the heterotic string in ref. [10] as required from the equivalence of the thermal cosmological constant and the free energy amplitude. As can readily be envisaged from eqs. (10) and (11), the “E-type” thermal amplitude L(b) is not modular invariant and annoyed with ultraviolet divergences at the endpoint t2A 0 for b G bH4

( 21k2) pka 8, where bH is the inverse Hagedorn temperature of the heterotic thermal

string. Existence of the limiting temperature b21H is in turn inherent to the exponential growth of the number of states when the mass goes to infinity. It is to be remembered that the thermal amplitude L(b) is infrared convergent at the upper limit t2K Q for any value of b which in turn reflects the absence of the tachyonic mode. We do not go into further details of the asymptotic behaviour of L(b) but merely refer to [10].

Let us now postulate the one-loop dual symmetric thermal cosmological constant L

–

(b ; D) at any space-time dimension D as an integral over the fundamental domain F,

i.e. 21/2 Gt1G 1 / 2 ; t2D 0 ; NtN D 1, of the modular group SL( 2 , Z) as follows [10]: L – ( b ; D) 4216 b ( 2 pa 8) 2D/ 2

!

(s , r)



F d2_t 2 pt22 B( t–, t ; D) Asr(t ; D) Dsr( t–, t ; b) , (12) where B t–_{, t ; D) 42}1 8( 2 pt2) 2(D 2 2 ) / 2_z–2(D 1 14 ) / 24_z2(D 2 2 ) / 24 3 (13) 3

k

1 1480

!

m 41 Q s7(m) z–m

l

»

n 41 Q ( 1 2z–n₎2D 2 14_{( 1 2z}n₎2D 1 2_,

u

A_{1 2}(t ; D) A_{2 1}(t ; D) A_{2 2}(t ; D)

v

4 8

g

p 4

h

(D 22)/6

u

2[u2( 0 , t) /u18 ( 0 , t) 1 /3 ](D 22 ) /2 2[u4( 0 , t) /u18 ( 0 , t)1 /3](D 22 )/ 2 [u3( 0 , t) /u18 ( 0 , t)1 /3](D 22 )/ 2

v

, (14) Dsr(t–, t ; b) 4Cs(1)(t–, t ; b) 1rCs(2)(t–, t ; b) , (15) Cs(g)(t–, t ; b) 4 (4p2a 8 t2)1 / 2

!

(p , q) exp

y

₂p 2

g

b2 2 p2_{a 8}p 2 1 2 p 2 a 8 b2 q 2

h

_t 21 ippqt1

z

, (16)

the signatures s , r and g read s , r 41, 2; 2, 1; 2, 2 and g41, 2, respect-ively, the summation over p[q] is restricted by (21)p

4 s[ (21 )q4 g] and the explicit use has been made of the Jacobi theta functions uj( 0 , t); j 41, 2, 3, 4 as well as the

Poisson resummation formula. It is almost needless to mention that the D 410 thermal amplitude b L–_{(b ; D 410) is literally reduced to the “D-type” representation of the} thermo-partition function Vh(b) in ref. [10] which in turn guarantees L

–

(b ; D 410) 4 L(b) as expected from self-consistency. Let us now examine the algebraic structure of the “D-type” thermal amplitude L–(b ; D) with the arithmetic aid of appendix B in ref. [10]. Typical theoretical observations are as follows: the scalar product

!

(s , r)AsrDsr is invariant under permutations of the signature, irrespective of the

values of b and D, not only for the shifting transformation t Kt11 but also for the inversion t K2t21_{. In addition, B(t}–_{, t ; D) is invariant, irrespective of the value of D,} under the action of any modular transformation. Accordingly, the thermal amplitude L

–

(b ; D) is manifestly modular invariant and free of ultraviolet divergences for any value of b and D. We have thus succeeded in regularizing the thermal amplitude L(b) à

la ref. [10] as well as ref. [13] through transforming the physical information in the

ultraviolet region of the half-strip E into the modular invariant integrand of the thermal amplitude L–(b ; D) over the fundamental domain F. As a matter of fact, moreover, the generalized duality symmetry [10] Cs(g)(t–, t ; b) 4Cg(s)(t–, t ; bA) holds for

any value of D, where bA_{4 2 p}2

a 8 /b. If and only if D410, on the other hand, the scalar

product

!

(s , r)AsrDsris invariant under the thermal duality transformation bD bA as a

simple and natural consequence of the Jacobi identity u422 u431 u444 0 for the theta functions. We are then led to conclude that the thermal duality relation b L–_{(b ; D) 4}

b

A L–(bA; D) is manifestly broken for the thermal amplitude L–(b ; D c 10 ) off the critical dimension.

It still remains for us to determine the infrared behaviour of the thermal amplitude L

–

(b ; D 410) in proper reference to the ultraviolet behaviour of the thermal amplitude L(b) which is in turn inherent to the exponential increase of the state density according as the mass grows to infinity. Let us recall that u8121 /3A ept2/12; u2A 0 ; u3A 1 ; u4A 1 near t2K Q. The infrared behaviour of the thermal cosmological constant L

–

(b ; D) is then asymptotically described at t2K Q as

(17) L–( b ; D) 4216 b ( 2 pa 8) 2D/ 2



F d2t 2 pt22 B(t–, t ; D) 3 3[A2 1(t ; D) 2A2 2(t ; D) ] C2(2)(t –_{, t ; b) ,} which is in turn paraphrased into the form

(18) L–_{(b ; D)42 64}k2( 8 p2_{a 8)}2D/ 2

!

(p , q)



21 O2 1 O2 dt1exp [ippqt1]

o

b A b



k12t21 Q dt2t22(D11)/ 23 3exp

y

2p 2 t2

u

b b Ap 2 1 b A b q 2 2 5 12(D 210)26

v

z

, where p , q 461; 63; 65; R . As can easily be seen from eq. (18), uniform convergence of the thermal amplitude L–(b ; D) is assured at any value of b if and only if

D E2/5. We are mainly interested in the case D410, anyhow. Infrared convergence of

the D 410 TFD amplitude L–(b ; D 410) is then guaranteed if and only if either ( 2 1k2) pka 84bHE b E Q or 0 E b E bAH4 ( 2 2k2) pka 8, where bAH reads the inverse dual Hagedorn temperature of the heterotic thermal string, i.e. there appear inevitably infrared divergences associated with the presence of the tachyonic mode for

b

A

HG b G bHin the modular invariant TFD amplitude L –

(b ; D 410) which are absent in the original TFD amplitude L(b). The present observation will be summarized à la ref. [13] as follows: firstly, the presence of the tachyonic mode is crucial to assure modular invariance of the TFD thermal string amplitude. Secondly, the resultant tachyonic divergence bears the dual relationship to the exponential growth of the state density as a function of the mass.

All we have to do is reduced to materializing the dimensional regularization of the dual symmetric TFD amplitude L–_{(b ; D 410). Explicit calculation of the t}2 integral in eq. (18) is readily performed for the case D E2/5 and yields

(19) L–_{(b ; D) 42}128 kp( 16 pa 8) 2D/ 2

!

(p , q)



21 O2 1 O2 dt1exp [ippqt1] 3 3

o

b A b

u

b b A p 2 1 b A bq 2 2 5 12(D 210)26

v

(D 21) /2 3 3G

y

2D 21 2 , p 2

k

1 2t1 2

u

b b Ap 2 1 b A b q 2 2 5 12(D 210)26

v

z

, irrespective of the value of b, where G is the incomplete gamma function of the second kind. We are now in the position to carry out the dimensional regularization in the sense of analytic continuation of the TFD amplitude (19). The right-hand side of eq. (19) indeed obeys the thermal duality symmetry b L–_{(b ; D) 4 b}A L–(bA; D) and brings forth the correct analytic continuation from D E2/5 to higher values of D, i.e. D410. We can therefore define the dimensionally regularized, D 410, one-loop dual symmetric thermal cosmological constant L×(b) by

(20) L×(b) 4 422 b( 8 pa 8) 2(D21 ) / 2

!

(p , q)



21 O2 1 O2 dt1exp [ippqt1]

g

b2 2 p2 a 8p 2 12 p 2 a 8 b2 q 2 262ie

h

(D21 ) /2 3 3G

y

2D 21 2 , p 2

k

1 2t1 2

g

b 2 2 p2_{a 8}p 2 1 2 p 2 a 8 b2 q 2 2 6 2 ie

h

z

, _{D 410 ,} which manifestly satisfies the thermal duality relation b L×(b) 4 bAL×(bA). As a matter of fact, the so-called ie prescription D 1ie is adopted à la ref. [3] as well as ref. [5] under explicit evaluation of the dimensionally regularized, TFD amplitude L×(b) in direct association with the nonvanishing decay rate of the tachyonic thermal vacuum. It must be emphasized that the present dimensional regularization based upon the TFD algorithm is in full accordance with the thermal stability of the fundamental properties such as modular invariance.

Let us next examine the singularity structure of the dimensionally regularized,

D 410 dual symmetric thermal amplitude L×(b). The position of the singularity

b_{NpN , NqN} is determined by solving b/ bA Qp2

1 bA /bQq22 6 4 0 for every allowed (p , q) in eq. (20). We then obtain two sets of solutions with NpqNG3 as follows: i) b1 , 14 bH4 (k_211)pk_{2 a 8; b}A1 , 14bAH4(k221)pk2 a 8, ii) b1 , 34bA3 , 14k3pk2 a 8; b3 , 14bA1 , 34 1 /k3 Q pk2 a 8. In particular, b1 , 1and bA1 , 1form the leading branch points of the square-root type at bH and bAH, respectively. Moverover, bH21 [bAH21] represents the lowest temperature singularity for the physical b [dual bA] channel. Both b1 , 3 and b3 , 1 are ordinary points and consequently left out of consideration. It is of practical importance to note that there exists no self-dual leading branch point at b04 bA04 pk2 a 8 as well as any non-leading branch point on the physical sheet of the inverse temperature. The present theoretical observation based upon the TFD free energy amplitude of the

D 410 heterotic thermal string yields a striking contrast to the previous argument by

ourselves [8] for the D 426 closed bosonic thermal string theory in the TFD framework. We are now in the position to touch upon the global phase structure of the

D 410 heterotic thermal string ensemble. Analysis is performed à la ref. [10, 15-17]

through the microcanonical ensemble paradigm outside the analyticity domain of the canonical ensemble. In particular, substantial use is made of the thermal duality symmetry not only for the canonical region but also for the microcanonical region. There will then appear three phases in the sense of the thermal duality symmetry as follows [8, 10, 15]: i) the b channel canonical phase in the domain ( 2 1k_{2) pa 84}

4 bHG b E Q, ii) the dual bA channel canonical phase in the domain 0 EbG bAH4 4 ( 2 2k2) pk_{a 8 and iii) the self-dual microcanonical phase in the domain b}AHE b E bH. In sharp contrast to the global phase structure of the D 426 closed bosonic thermal string ensemble [8], however, there will occur no effective splitting of the microcanonical region because of the absence of the self-dual branch point at b04 bA04 pk2 a 8 as well as any secondary singularity. As a consequence of the self-duality of the microcanonical phase, therefore, it may be possible to claim that the so-called maximum temperature of the D 410 heterotic-string excitation is asymptotically described at least at the one-loop level as bAH21[b21H ] in replacement of b0214 bA021 for the physical b [dual bA] channel. It seems almost needless to mention that the fruitful

thermodynamical investigation of string excitations will be prerequisite for the real solid substantiation of the present novel hypothesis on the “true” maximum temperature in proper reference to the global phase structure of the D 410 heterotic thermal string ensemble. It is hoped that we can shed some light upon this unfathomable subject with the new-fashioned aid of the D-brane paradigm [18] in a future communication.

* * *

One of the authors (HF) is grateful to Prof. S. SAITO for the hospitality of Tokyo

Metropolitan University.

R E F E R E N C E S

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[18] For a nearly related publication, see VA´ZQUEZ-MOZO M. A., Princeton preprint IASSNS-HEP-96-73; hep-th/9607052 (1996).