On running couplings in gauge theories from type-IIB supergravity

20 May 1999

Physics Letters B 454 Ž1999. 270–276

On running couplings in gauge theories from type-IIB supergravity A. Kehagias 1, K. Sfetsos

2

Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland Received 26 February 1999 Editor: L. Alvarez-Gaume´

Abstract We construct an explicit solution of type-IIB supergravity describing the strong coupling regime of a non-supersymmetric gauge theory. The latter has a running coupling with an ultraviolet stable fixed point corresponding to the N s 4 SUŽ N . super-Yang–Mills theory at large N. The running coupling has a power law behaviour, argued to be universal, that is consistent with holography. Around the critical point, our solution defines an asymptotic expansion for the gauge coupling beta-function. We also calculate the first correction to the Coulombic quark–antiquark potential. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction and computations One of the well-known vacua of the type-IIB supergravity theory is the AdS5 = S 5 one, first described in w1x. The non-vanishing fields are the metric and the anti-self-dual five-form F5 . The latter is given by the Freund–Rubin-type ansatz, which is explicitly written as

'L Fmnrk l s y

2

emnrk l ,

m , n , . . . s 0,1, . . . ,4 ,

'L e , i , j, . . . s 5, . . . ,9 , Ž 1. 2 i jk p q and is clearly anti-self-dual. This background has received a lot of attention recently because of its Fi jk p q s

1 2

E-mail: [email protected] E-mail: [email protected]

conjectured connection to N s 4 SUŽ N . superYang–Mills ŽSYM. theory at large N w2,3x. The SYM coupling g YM is given, in terms of the dilaton 2 F , as g YM s 4p eF , and the ’t Hooft coupling is 2 2 g H s g YM N, where N s HS 5 F5 is the flux of the five-form through the S 5. The dilaton is constant in this background, which is related to the finiteness of the N s 4 SYM theory. In order to make contact with QCD, it is important to investigate deformations of the SYM theory that break conformal invariance and supersymmetry. In this case, the couplings are running corresponding to a non-constant dilaton in the supergravity side. It is then clear that the background we are after is a perturbation of AdS5 = S 5. Attempts to find supergravity backgrounds that allow a non-constant dilaton, and hence a running coupling of the YM theory, have been exploited within type-0 theories w4–6x.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 9 3 - 7

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

Deformations of the N s 4 theory, which flow to an interacting conformal fixed point, have been considered in w7x. The purpose of the present paper is to show that running couplings are also possible within the typeIIB string theory. We will study the ‘‘minimal’’ case, that is we will keep the same F5 form as in Ž1. and turn on a non-constant dilaton. We will show that such a solution, which breaks supersymmetry and conformal invariance, exists. We will assume for the metric four-dimensional Poincare´ invariance ISO Ž1,3., since we would like a gauge theory defined on Minkowski space–time. In addition, we will preserve the original SO Ž6. symmetry of the AdS5 = S 5. As a result, the ISO Ž1,3. = SO Ž6. invariant tendimensional metric is of the form 3

271

Moreover, the non-zero components of the Ricci tensor for the metric Ž2. are Rrr s y4Er2 ln V , R ab s yha b Er2 ln V q 3 Ž Er ln V .

ž

2

/,

Ž 6.

and the first equation in Ž3. reduces to solving

Ž Er ln V .

2

A2 s 24

Vy6 q

L 4

V2 .

Ž 7.

The solution of the above equation for V as a function of r is given implicitly, in terms of a hypergeometric function, by

V3 F

ž

6L

1 3 11 2 8 8

, , ;y

A2

(

/

V 8 s"

3 8

< A<Ž r y r0 . ,

Ž 8.

ds 2 s gmn dx mdx n q g i j dx i dx j , where gmn dx mdx n s V 2 Ž r . Ž d r 2 q dx a dx a . ,

a s 0,1,2,3 ,

Ž 2. 5

and g i j is the metric on S . The dilaton, by ISO Ž1,3. = SO Ž6. invariance, can only be a function of r . The supergravity equations turn out to be Rmn s yL gmn q 12 EmFEn F , 1

mn ' 'y G Em Ž y G G EnF . s 0 ,

Ž 3.

where r 0 is another constant of integration. The different overall signs in the right-hand side of Ž8. arise from taking the square root in Ž7.. We impose the boundary condition that the space described by Ž2. becomes AdS5 when r ™ 0q. That means that the conformal factor should assume the form V , Rrr for small r , where R ' Ž4p g s N .1r4 . Using well-known formulae for the hypergeometric functions, we see that this naturally leads to the choice of the minus sign in Ž8. and also fixes L s 4rR 2 . In addition, the constants A and r 0 are related by < A< s R3

and Ri j s L gi j .

Ž 4.

The above equation is automatically solved for a five-sphere of radius 2r 'L and a first integral of the dilaton equation in Ž3. is

ErF s A Vy3 ,

h'

(

where A is a dimensionful integration constant.

,

r 04 3

G Ž 3r8 . G Ž 1r8 .

8p

24 3r8

, 1.87 .

Ž 9.

Then Ž8. can be written as 3

Vr 0

ž / ž ( ž F

R

Ž 5.

h4

s

3 8

1 3 11 24 , , ;y 8 2 8 8 h

h4 1y

r r0

/

8

Vr 0

ž // R

.

Ž 10 .

We also find that

3

Supersymmetric solutions to type-IIB supergravity which, however, do not preserve the Poincare´ invariance in the brane world-volume have been found in w8x.

V , Ž 3r8 .

1r6

h 4r3

R

r0

ž

r 1y

r0

1r3

/

,

as

r ™ ry 0.

Ž 11 .

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

272

We may solve the dilaton equation in Ž5. close to r s 0 and r s r 0 . The result for the string coupling is

ž

eF s g s 1 q s

h4

11h 12 qs 3456

4

r

h8

r

32

r0

ž / ž / ž / /

8

q

r0

4

12

r

q... ,

r0

as

r ™ 0q,

Ž 12 . 4

where s s "1 s signŽ A.,

F s F 0 y s Ž 8r3 .

1r2

while on the other side

r

ž

ln 1 y

r0

/

,

as

r ™ ry 0.

Ž 13 . The form of the above solution is dictated by the fact that, in the limit r 0 ™ `, the dilaton should be eF s g s ' eF 0 and Ž12. and Ž13. should coincide. Now, at r s r 0 there is a singularity that may be easily seen by computing the Ricci scalar using Ž3.. The latter is R s 12 EmFE mF , which at r , r 0 behaves as R ; Ž r 0 y r .y8 r3. Hence, we may consider our solution only as an asymptotic expansion around the AdS5 geometry at r s 0 of the form

VŽ r. s

R

r

`

ž

1q

Ý ns1

8n

r

an

ž // r0

,

r - r0 ,

Ž 14 .

where the coefficients a n are computed using the series representation for the hypergeometric function in Ž10. for large V . The first coefficient of the expansion turns out to be a1 s yh 8r432 , y0.352. Hence, V Ž r . is given by

VŽ r. ,

R

r

ž

h8 1y

432

r

8

ž // r0

Ž 15 .

to a very good approximation, except when r takes values very close to r 0 . Then we may use, to a very

Fig. 1. Plot of V Ž r .r R in units where r 0 s1. Curves Ž1. and Ž2. were plotted using Ž15. and Ž11. respectively. The curve corresponding to V Ž r .r R, obtained by numerically solving Ž8., coincides with the union of these curves.

good approximation as well, Ž11. instead of Ž15., and the results are plotted in Fig. 1. Note that our analysis was done in the Einstein frame and it is not difficult to translate everything into the string frame by multiplying the Einstein metric by eF r2 . Our solution is singular at r s r 0 in both frames and can, indeed, be trusted away from that point. Note that, the metric Ž2. with the conformal factor V specified by Ž10. can be written in horospherical coordinates 5 Ž r, x a . determined by V Ž r . d r s dr, as gmn dx mdx n s dr 2 q K Ž r . dxa dx a , where K Ž r . s '2 R 2 ey2 r 0 r R sinh1r2

ž ž

e s g s coth

The solutions corresponding to the two possible choices for s are related by an S-duality transformation and correspond to different gauge theories. This reflects the fact that, except for r 0 s`, corresponding to the N s 4 SUŽ N . SYM at large N, our solution describes gauge theories that are not S-duality-invariant. Also in Ž12. we have used Ž14. below.

ž

4Ž r0 y r . R

/

.

Ž 17 .

In the same coordinate system the string coupling takes the form F

4

Ž 16 .

2Ž r0 y r . R

2 sa

//

.

a ' '6 r4 . Ž 18 .

In the rest of the paper we prefer to work in the Ž r , x a . coordinate system.

5

We thank A.A. Tseytlin for comments on this point.

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

273

2. Running coupling In the AdSrCFT scheme, the dependence of the bulk fields on the radial coordinate r may be interpreted as energy dependence. In fact, it is a general feature in the AdSrCFT scheme that long Žshort. distances in the AdS space correspond to high Žlow. energies in the CFT w2,9x. In particular, if the dilaton in the supergravity side is a function of r , then the ’t Hooft coupling of the boundary CFT has an energy dependence and can be interpreted as the running coupling of the CFT. Running coupling means of course that we are away from conformality; thus, backgrounds that admit non-constant dilaton correspond to non-conformal field theories. As long as supersymmetry is unbroken, spin-sum rules for the AdS supersymmetry are expected to protect the ’t Hooft coupling g H of the boundary N s 4 YM theory against running. However, if supersymmetry is broken, there are then no more cancellations between fermionic and bosonic contributions leading to the running of g H . The specific background we found here clearly breaks supersymmetry, and

Fig. 2. Plot of g H r g H) as a function of U using Ž20..

of the beta-function for the ’t Hooft coupling around g H) is U

dg H dU

s y4 Ž

g H y g H)

. y2

14

Ž g H y g H) .

27

g H) 2

y

Ž g H y g H) .

2

g H)

3 4

q O Ž g H y g H) . .

Ž 21 . dl s 12 g ME M Fe ) s

A 2V 3

g re ) ,

dc M s DM e

Ž 19 . are the associated fermionic zero modes. If we now follow the correspondence between longdistancesrhigh-energies in the AdSrCFT scheme, we find that the dual theory of the supergravity solution we obtained has a coupling with power-law running. Indeed, by changing the variable r s R 2rU and interpreting U as the energy of the boundary field theory, we find from Ž12. the running of g H :

h4

ž

g H s g H) 1 q s

R8

8 r 04 U 4

17h 12

R 24

27648

r 012 U 12

qs g H) s R 2

h8 q

R 16

128 r 08 U 8

However, the above equation does not specify the beta-function, but rather its derivative at the g H s g H) point. The reason is that our solution breaks down at energies U ; R 2rr 0 . From Ž21. we see that

b X Ž g H) . s y4 ,

Ž 22 .

which means that is a UV-stable fixed point. 6 We believe that Ž22. is universal, namely, that it is valid for all models that approach AdS5 = S 5 at some boundary. This can be seen by recalling that, near AdS5 = S 5, the dilaton always satisfies Ž3. with V s Rrr . As a result, F will behave as eFy F 0 ; r 4 , where F 0 is the value of F at r s 0. We also see that Ž21. determines the second and third derivatives of the coupling beta-function at the fixed point, ) g YM

6

/

q... ,

Ž 20.

where is the UV value of the ’t Hooft coupling Žthe result in plotted in Fig. 2.. From this expression, it may easily be found that the behaviour

Using a radiousrenergy relation in horospherical coordinates of the form Us R 2 eyr r R we find that the running of g H is 1

U

dg H dU

sy ag H)

gH

žž / g H)

1q

1

a

y

gH

ž / g H)

1y

a

/

.

Ž 23.

However, the above expression is trustable only around the fixed point g H) .

274

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

which, however, are not expected to be model-independent. Let us also note that there is no known perturbative field theory with UV-stable fixed points. A behaviour of the form Ž20., namely power-law running of the couplings, was also found in type-0 theories Žsee second ref. in w6x., in gauge theories in higher dimensions w10x and extensively discussed in gauge-coupling unification in theories with large internal dimensions w11–14x. In this scenario, the internal dimensions are shown up in the four-dimensional theory as the massive KK modes. These modes can run in the loops of the four-dimensional theory, giving rise to a power law running of the couplings. In particular, for d large extra dimensions and for energies E above the infrared cutoff, which is specified by the mass scale m of the extra dimensions, we find, just for dimensional reasons, that the running coupling constant of the effective four-dimensional theory is of the form g Ž4. y g 0Ž4. ; Ž mrE . d r2 , where g 0Ž4. is the bare coupling. Thus, in our case, since we have a four-dimensional theory coming from ten dimensions, we should expect the coupling to run in the sixth power of E. Instead, we find here that the coupling depends on the fourth power of U s E, indicating that when holography is involved, we get a softer running of the couplings. It is possible to identify those operators that are responsible for the running of the coupling in the boundary field theory. Since the dilaton approaches a constant value at r s 0 and the asymptotic background is an AdS5 space, the corresponding boundary field theory is expected to be a deformation of the N s 4 SUŽ N . supersymmetric YM theory. The explicit form of the deformation may be specified by recalling that our solution still has an SO Ž6. symmetry. There are not many SO Ž6. singlets in the spectrum of the S 5 compactification. In fact, from the results in w15x we see that the only scalar singlets are the complex scalar of type-IIB theory B, a i jk l and h ii , with masses m2B s 0, m 2a s m2h s 32 in AdS-mass units. Since we have not perturbed the five-form F5 , a i jk l s 0 and thus the perturbations we have turned on are the real part of B and h ii . From their masses we find that the former corresponds to marginal deformations of the type Fmn2 , while the latter corre4 sponds, to the dimension-eight operator Fmn , which is irrelevant. However, it gives contributions to the boundary field theory since we have an IR cutoff

specified by r 0 . Sending r 0 ™ ` all bulk perturbations disappear and, similarly, the boundary field theory turns out to be the N s 4 large-N SUŽ N . SYM theory.

3. The quark–antiquark potential The breaking of the superconformal invariance of the N s 4 theory by our solution should be apparent in the expression for the quark–antiquark potential, which we now compute along the lines of w16,17x. We will find corrections to the purely Coulombic behaviour, which, on purely dimensional grounds, we expect to be in powers of AL4 , where L is the quark–antiquark distance. We are eventually interested in the first such correction, which, as is apparent when comparing Ž14. with Ž12., is due to the dilaton, but for the moment we keep the formalism general. As usual, we have to minimize the Nambu–Goto action 1 Ss dt d s det Ž GM N Ea X MEb X N . , Ž 24 . 2p where GM N is the target-space metric in the string frame. For the static configuration x 0 s t , x 1 ' x s s , and x 2 , x 3 as well as the coordinates of S 5 held fixed, we find that Ž24. becomes Žwe use the notation of w16x. T 2 Ss dx e Q r2 Ž Ex U . q U 4rR 4 , Ž 25 . 2p

(

H

(

H

2

8n

where e Q ' eFS 4 and S s 1 q Ý`ns 1 a n rR0 U is the function multiplying Rrr in Ž14. Žrewritten using r s R 2rU .. It is clear that any background that approaches AdS5 will always have a S of the form Ž14., so that our analysis is quite general at this point. It is easy to see that the solution is expressed as

ž /

xs

U

R 2 dU

0

U 2 e Q y Q 0 U 4rU04 y 1

HU

(

,

Ž 26 .

where U0 is the smallest ‘‘distance’’ of the trajectory to the center and Q 0 is the value of the function Q evaluated at U0 . We assume that one of the branes is taken out to U s ` and that the string configuration starts and ends at this brane. The rest of the branes are located at U s R 2rr 0 . Setting x s Lr2 corre-

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

sponds to U s `. In turn, this gives a condition that relates U0 to L as R 2 dU

`

Ls2

HU

0

(

U 2 e Q y Q 0 U 4rU04 y 1

.

Ž 27 .

Proceeding in a standard fashion, we substitute back our solution Ž26. into the action Ž25. and obtain an integral that is infinity. This is because we have included into the potential energy the Žequal. masses of the infinitely heavy Žin the supergravity approximation. quark and antiquark. In order to compute these masses, we assume that N-branes are at U s R 2rr 0 and 1 at U s Umax that is assumed to be large but finite. The mass of a single quark is computed if in Ž25. we consider a configuration with x 0 s t , U s s and with fixed spatial world-volume coordinates x a , a s 1,2,3, as well as S 5 coordinates. Then the self-energy of the quark is 1 Umax Eself s dUeQ r2 . Ž 28 . 2p R 2r r 0

H

Subtracting off this energy twice and letting Umax ™ `, we obtain a finite result for the quark–antiquark potential given by Eq q s

1

U 2rU04 e Q r2

`

H dU (U rU p U

2 Q 0y Q 0 ye

4

0

1

y

`

H dUe p R rr

Q r2

.

Ž 29 .

2

0

At this point we have to solve Ž27. for U0 as a function of L and substitute the result back into Ž29. to obtain Eq q as a function of L only. This can be done perturbatively in powers of Lrr 0 , and we are interested in the first correction to the Coulombic law behaviour of the potential. As explained, the correction due to the non-constant dilaton is dominant and we will therefore use V s Rrr , which corresponds to AdS5 = S 5 for the string metric, whereas for the string coupling we will keep the first two terms in Ž12.. Then we find that 2

U0 ,

h1 s

2 R h1 L

ž

1q

p 1r2G Ž 3r4 . G Ž 1r4 .

s

h

2

L

4

ž /ž //

8 2h1

, 0.599 ,

r0

,

Ž 30 .

275

where the value of the numerical constant h has been given in Ž9. and Eself ,

Umax

R2

ž

y

2p

2pr 0

h4 1ys

24

/

.

Ž 31 .

The result for the quark–antiquark potential is Eq q , y

2h12 R 2 1

p R2

q

pr 0

L

ž

4

h

s

4

L

ž ž /ž // 1y

h4 1ys

24

8 2h1

/

r0

.

Ž 32 .

We see that the Coulombic potential receives a correction proportional to L3 due to the breaking of conformal invariance. 7 This is remarkably similar to the potential obtained in w18x for the quark–antiquark pair for N s 4, at finite temperature, using the near-horizon supergravity solution for N coincident D3-branes. However, in that case supersymmetry is broken by thermal effects, whereas in our case it is broken, by the presence of a non-trivial dilaton, even at zero temperature. The last term in Ž32. does not depend on L and represents a constant shift of the potential energy. As a final remark we note that the computation of the potential for the monopole–antimonopole pair proceeds along the same lines as that for the quark– antiquark pair, with the only difference that we start, similarly to w19x, with the action for a D-string. This means that the integrand in Ž25. should be multiplied by eyF . Hence, the function e Q entering into Ž25. is defined as e Q s eyFS 4 . Consequently, the first correction to the Coulombic behaviour of the monopole–antimonopole potential is given by Em m , y

2h12 R 2

p gs L R2

q

pr 0

ž

h

s

4

L

4

ž ž /ž // 1q

8 2h1

h4 1qs

24

/

,

r0

Ž 33 .

which is the same as Ž32. after we use the fact that under S-duality g s ™ 1rg s and s ™ ys. Hence we

7 Supergravity is valid when r < r 0 , which means that U G U0 4 R 2 r r 0 . Using the leading term in Ž27., we deduce that Lr r 0 <1, which is indeed the condition for the validity of Ž32..

276

A. Kehagias, K. Sfetsosr Physics Letters B 454 (1999) 270–276

see a screening Žantiscreening. of the quark–antiquark pair for s s q1Žy1. and exactly the opposite behaviour for the monopole–antimonopole pair.

w7x

Acknowledgements We would like to thank K. Dienes, E. Dudas, T. Gherghetta, A.A. Tseytlin and A. Zaffaroni for discussions.

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