Page 1

arXiv:hep-th/0205136v3 13 Jun 2004

OHSTPY-HEP-T-02-006

hep-th/0205136

Scalar propagator in the pp-wave geometry obtained

from AdS5× S5

Samir D. Mathur1, Ashish Saxena2and Yogesh K. Srivastava3

Department of Physics,

The Ohio State University,

Columbus, OH 43210, USA

Abstract

We compute the propagator for massless and massive scalar fields in the metric

of the pp-wave. The retarded propagator for the massless field is found to stay

confined to the surface formed by null geodesics. The algebraic form of the massive

propagator is found to be related in a simple way to the form of the propagator in

flat spacetime.

1mathur@pacific.mps.ohio-state.edu

2ashish@pacific.mps.ohio-state.edu

3yogesh@pacific.mps.ohio-state.edu

1

Page 2

1Introduction.

Recently there has been a surge of interest in the study of string theory on pp-wave

backgrounds. A 10-d pp-wave metric (with a certain background 5-form field strength)

can be obtained as a limit of AdS5× S5: it is the metric seen by particles that rotate

fast around the diameter of the S5[1].

String states in such backgrounds were studied in [2]. To proceed further and study

interactions between such states one needs to know the propagators of the fields that are

exchanged between string states. Since we are in Type IIB string theory, the massless

bosonic fields that can be exchanged are the dilaton Φ and axion C, the NS-NS and RR

2-form gauge fields BNSNS

µν

and BRR

In this letter we compute the propagator for massless scalar fields like Φ, as well as for

massive scalar fields.

We note that the propagator in AdS space is known [3] , and in principle one can try to

extract the propagator in the pp-wave by starting with the propagators in AdS space for

different harmonics on the sphere, putting together these propagators in an appropriate

way and taking an appropriate limit. This process appears to be cumbersome though,

and we find it much easier to work directly with the pp-wave background instead.

Our principal results are the following.

propagators for the scalar, both for the massless field and the massive field. The retarded

propagator is found to stay confined to the surface formed by the null geodesics. For

both the massless and the massive propagators we find algebraic forms that are closely

related to the forms found in flat space. The results thus suggest that even for higher

spin fields there might be a simple way to guess the pp-wave propagators from flat space

computations.

One of our goals in finding the exact form of the propagator was to study the behavior

of highly excited string states which can collapse into black holes under their self-gravity.

We comment briefly on this problem in the discussion. While this paper was in prepa-

ration there appeared [4] which studied a similar question using an approximate form of

the propagator.

µν, the 4-form gauge field Aµνλρand the graviton hµν.

We compute the retarded and Feynman

2Constructing the propagator from solutions of the

wave equation

We consider the 10-D metric obtained from the pp-wave limit of AdS5× S5. The line

element in pp-wave background is

ds2= −4dx+dx−− r2(dx+)2+ dxidxi

(1)

2

Page 3

where xi,i = 1...8 are the transverse coordinates and r2= xixiis the distance squared

in the transverse coordinates. The scalar laplacian is

∇2

=

1

√−g∇a(gab√−g∇b)

r2

4∂2

=

−− ∂−∂++ ∇2

8

(2)

where ∇2

8= ∂i∂i. The wave-equation for mass m is

(∇2− m2)F = 0(3)

To find the propagator we will solve the eigenvalue problem for the above operator. Let

Fλbe an eigenfunction with eigenvalue −λ:

(∇2− m2)Fλ= −λFλ

We write

Fλ[x+,x−,xi] = Neik+x++ik−x−f[xi]

(N is a normalization) obtaining

8f −k2

4

(4)

(5)

∇2

−r2

f + (k+k−− m2+ λ)f = 0(6)

This is just the equation for a non-relativistic, spherically symmetric quantum harmonic

oscillator in 8 dimensions with frequency

ω =|k−|

2

(7)

and total energy

E =(k+k−− m2+ λ)

2

.(8)

Writing

f =

?

i

fi

(9)

we get

1

fi

d2fi

dx2

i

− ω2x2

i= −gi

(10)

with

Σ8

i=1gi= 2E. (11)

The normalised solutions are

fi=

?

√ω

2nini!√π

?1

2

Hni(˜ xi)e−1

2˜ x2

i

(12)

3

Page 4

where ˜ xi=√ωxi, Hnare the Hermite polynomials of order n and gi= 2(ni+1

can now write the normalized solutions to (4) as

2)ω. We

Fλ[x+,x−,xi] =

1

√2

1

2πeik+x+eik−x−Π8

i=1

?

√ω

2nini!√π

?1

2

Hni(˜ xi)e−1

2˜ x2

i

(13)

with

λ = −[k+k−− |k−|Σ8

i=1(ni+1

2) − m2](14)

The propagator satisfying

(△ − m2)G(x2,x1) =

1

√−gδ(x2− x1)(15)

is then given by (√−g = 2)

G(x2,x1) =

?

?∞

λ

Fλ(x2)F∗

(−λ)

dk+dk−

2(2π)2eik+(x+

λ(x1)

=

−∞

2−x+

1)eik−(x−

2−x−

1)

Σ{ni}Π8

i=1

?

√ω

2nini!√π

?

Hni(˜ x1i)e−1

k+k−− |k−|Σ8

2˜ x2

1iHni(˜ x2i)e−1

i=1(ni+1

2˜ x2

2i

2) − m2

(16)

3Simplifying the expression for the propagator

Since we encounter the absolute value of k−in ω eq. (7), it is convenient to break up the

k−integral into two parts: a part (−∞,0) and a part (0,∞). We write n ≡ Σ8

∆x±= x±

i=1niand

2− x±

1. Then

G(x2,x1) = I++ I−

(17)

where

I+= Σ{ni}1

2

1

(2π)6

?∞

0

dk−k4

−eik−∆x−

k−

1

2nini!Hni(√ωx1i)Hni(√ωx2i)e−ω

?∞

−∞dk+

eik+∆x+

k+− (n + 4) − m2/k−

Π8

i=1

2(x2

1i+x2

2i)

(18)

and after a change of variables k−→ −k−we can write I−as

I−= −Σ{ni}1

2

1

(2π)6

?∞

0

dk−k4

−e−ik−∆x−

k−

1

2nini!Hni(√ωx1i)Hni(√ωx2i)e−ω

?∞

−∞dk+

eik+∆x+

k++ (n + 4) + m2/k−

Π8

i=1

2(x2

1i+x2

2i)

(19)

4

Page 5

Consider any point in the pp-wave spacetime and look at the light-cone in the in-

finitesimal vicinity of this point. First ignore the xi, and look at the light cone in the 2-D

space x+,x−. The two lines forming this cone are x+= 0, and x−= −

vector Vµwhich has V+> 0 (and all other components zero) is timelike in the metric

(1); this tells us which of the four sectors marked out by the two null lines is the forward

light cone. We find that at least for the infinitesimal vicinity of the starting point a

retarded propagator can be defined by requiring that the propagator be nonzero only for

∆x+> 0. We adopt this as our definition of retarded propagator (we will return to a

more complete discussion of null geodesics later).

Let us begin with I+and perform the integral over k+. For ∆x+> 0 we can close the

k+contour in the upper half plane, so to get a nonvanishing result from each pole we shift

the poles to slightly above the real axis. The k+integral just sets k+= (n+4)+m2/k−.

Let us define

z = ei∆x+

x+

4xixi. Further, the

(20)

For any one choice of the index i the sum in Σ{ni}over the Hermite polynomials can be

evaluated by using the identity [7]

Σ∞

ni=0

Hni(√ωx1i)Hni(√ωx2i)(z

2)ni

ni!

=

1

√1 − z2e

ω

?

2x1ix2iz−x2

1iz2−x2

1−z2

2iz2

?

(21)

We need to take a product over 8 values of the index i. We then find

I+=

2πi

(2π)6

z4

(1 − z2)4

Θ(∆x+)

2

?∞

0

dk−k3

−e−ik−Y2eim2∆x+

k−

(22)

where

Y2= −∆x−+x2

1+ x2

4i

2

+x2

1z2+ x2

2z2− 2x1.x2z

2i(1 − z2)

Before proceeding further with the evaluation of I+we take note of the regularizations

required to define I+. It is a general feature of Green’s functions that for sufficiently

high spacetime dimensionality the Green’s function is not square integrable, and so the

Fourier transform need not be given by a convergent integral – a damping factor must be

introduced to cut off high frequencies. Lifting this cutoff at the end brings the Green’s

function back to its required form as a singular ‘distribution’.

The damping at high frequencies can be provided by introducing into I+,I− the

factors:

e−ǫ|k−|e−ǫ′ni

(23)

The first of these factors leads to the change in (22)

Y2→ Y2− iǫ (24)

The second factor in (23) leads to the change that in the identity (21) we have the

replacement

z → ze−ǫ′

(25)

5

Page 6

With these regulations we find

I+=Θ(∆x+)

2

2πi

(2π)6

1

(2sin(∆x++ iǫ′))4

?∞

0

dk−k3

−e−i(k−(Y2−iǫ)−m2∆x+

k−

)

(26)

and

Y2=

Φ

4sin(∆x++ iǫ)

(27)

where

Φ = −4∆x−sin(∆x++ iǫ′) − 2sin2(∆x++ iǫ′)

2

(x2

1+ x2

2) + (x1− x2)2

(28)

is an expression that goes over to the invariant distance squared for small separations.

Proceeding similarly for I−, we require again that the contribution be nonzero only

for ∆x+> 0, and we get

I−= −Θ(∆x+)

Note that I−is precisely the complex conjugate of I+.

2

2πi

(2π)6

1

(2sin(∆x+− iǫ′))4

?∞

0

dk−k3

−ei(k−(Y2+iǫ)−m2∆x+

k−

)

(29)

4Closed form expressions for the propagators

4.1 The massless propagator

Let us first work out the case m2= 0. From (26) we find that

I+=Θ(∆x+)

2

2πi

(2π)6

1

(2sin(∆x++ iǫ′))4

6

(Y2− iǫ)4= 3Θ(∆x+)2πi

(2π)6

24

(Φ − i˜ ǫ)4

(30)

where

˜ ǫ = (4sin∆x+)ǫ(31)

Similarily,

I++ I−= 3Θ(∆x+)2πi

(2π)6

?

24

(Φ − i˜ ǫ)4−

24

(Φ + i˜ ǫ)4

?

(32)

Noting that

1

(Φ − i˜ ǫ)−

1

(Φ + i˜ ǫ)= 2πi (sign ˜ ǫ)δ(Φ)(33)

we get

Gret(x2,x1) = Θ(∆x+)1

2π4

sin∆x+

|sin∆x+|δ

′′′(Φ) (34)

where the derivatives of the delta function are with respect to its argument Φ.

It is readily verified that this propagator has the correct normalization to agree at

short distances with the retarded propagator satisfying (15) (with m2= 0).

6

Page 7

4.2The massive propagator

Let us now consider the expression (22) for I+for the massive scalar field. Performing

the integral we find [6]

I+= −π2

16Θ(∆x+)

1

(2π)6

m4(∆x+)2

(Y2− iǫ)2sin4(∆x++ iǫ′)H(1)

−4(2m

?

−(Y2− iǫ)

√

∆x+) (35)

where H(1)is the Hankel function of the first kind. For I−we get the complex conjugate

of I+

I−=π2

16Θ(∆x+)

1

(2π)6

m4(∆x+)2

(Y2+ iǫ)2sin4(∆x+− iǫ′)H(1)

−4(−2m

?

−(Y2+ iǫ)√∆x+)(36)

Note that the Hankel function is H(1)

logarithmic branch cut at z = 0, so in principle we must decide which branch of the

Hankel function we are on. With the given regulations in (22) the integral is well defined

for all values of ∆x+> 0,m2> 0, which suggests that there should be no ambiguity in

the choice of branch, and that we should be on the principal branch which is defined by

the integral.

To observe that this is indeed what happens, we look at the behavior of the argument

of the Hankel function when ∆x+passes through nπ (which causes sin∆x+to vanish)

or when we enter or leave the light cone. We find that with the given regulations the

argument of the Hankel function in I+stays (for all values of the coordinates) in the

first quadrant of the complex plane. Since the argument does not thus circle the origin,

we stay on the same branch of the Hankel function. This branch is given by continuing

the value of H(1)for real positive arguments to the first quadrant of the complex plane.

A similar analysis holds for I−, and so no ambiguity exists in the value of the retarded

Green’s function which is

Gret= I++ I−

−4= J−4+ iN−4. The function N−4(z) has a

(37)

with I+,I−given by (35), (36).

We observe that as expected the choice of branch indicated by the regularizations

does not depend on the ratio of the two regularizations ǫ,ǫ′, and from now on we just

write ǫ for both.

4.3 The Feynman propagator

To define the analogue of the flat space Feynman propagator we require that positive

frequencies in the variable x+travel forward in x+and negative frequencies travel back-

wards in x+. To make I+nonzero at negative ∆x+we must shift the poles in k+integral

in (18) into the lower half plane. We then get instead of (35) the result

˜I+=π2

16Θ(−∆x+)

1

(2π)6

m4(∆x+)2

(Y2− iǫ)2sin4(∆x++ iǫ′)H(1)

−4(2m

?

−(Y2− iǫ)

√

∆x+)(38)

7

Page 8

The Feynman propagator is then

GF(x2,x1) =˜I++ I−

(39)

For completeness we record the result of shifting poles in I− such that we get a

contribution only for ∆x+< 0:

˜I−= −π2

16Θ(−∆x+)

1

(2π)6

m4(∆x+)2

(Y2+ iǫ)2sin4(∆x+− iǫ)H(1)

−4(−2m

?

−(Y2+ iǫ)

√

∆x+)

(40)

5Similarities with flat space propagators

5.1The massless propagator

If spacetime is flat and its total dimension is even then the retarded propagator G(x2,x1)

for massless fields stays confined to the null cone, which is the cone generated by the

tracks of null geodesics starting at x1. We will now observe that a similar property holds

for the retarded massless propagator in the pp-wave.

Let us first compute the geodesics in the pp-wave metric. If τ is an affine parameter

along the geodesic, then the geodesic equations are

d2x+

dτ2= 0(41)

d2x−

dτ2+ ˙ x+

8

?

i=1

xi˙ xi= 0(42)

d2xi

dτ2+ (˙ x+)2xi= 0(43)

where a dot denotes the derivative with respect to τ. The first equation gives

x+= ατ + β(44)

If α ?= 0, then we can scale τ to set x+= τ. (For massive geodesics τ will not be the

proper distance along the geodesic but rather a multiple of the proper distance.) For the

initial conditions

x−(τ = 0) = x−

dx−

dτ(τ = 0) =

xi(τ = 0) = x1i

dxi

dτ

1

˙ x−

1

=˙ x1i

(45)

8

Page 9

we get the solution

x−(τ) − x−

1

=

˙? x1

2− ? x1

8

˙? x1sin τ + ? x1cos τ

2

sin 2τ +

˙? x1.x1

4

cos 2τ + Cτ −

˙? x1.x1

4

(46)

? x(τ) = (47)

where ? x denotes the coordinates in the transverse 8 dimensional space, and

C = ˙ x−

1−

˙? x

2

1− ? x2

4

1

(48)

For null geodesics the initial conditions imply that C = 0. A little algebra then shows

that along such geodesics

Φ(x,x1) = 0(49)

so that the retarded propagator (34) is confined to the surface formed by the null

geodesics.4

In flat space the propagator is a function of the invariant distance squared

s2

flat= −4∆x−∆x++ (? x1− ? x2)2. (50)

We have

Gflat

ret(x2,x1) = Θ(t)

1

2(π)4δ

′′′(−s2) (51)

We thus see that the pp-wave result (34) is related to the flat space result by the

replacement (−s2

flat) → Φ.

5.2The massive propagator

For ∆x+<< 1 we can replace sin∆x+→ ∆x+and the propagator reduces to the

propagator in flat space, as expected. What is interesting is that the algebraic form of

the massive propagator in the pp-wave is closely related to the form of the propagator

in flat space.

For flat space

Iflat

+

= −π2

16Θ(∆x+)

1

(2π)6

16m4

(−s2

flat)2H(1)

−4(m

?

−s2

flat) (52)

with similar expressions for the other functions involved in the propagators. Now note

that for a timelike geodesic

ds2

(dx+)2= −4C(53)

4If α = 0 in (44) then we have the solution x+= x+

cannot be timelike, and the only null geodesic is (after scaling τ) x+= const., x−= τ, ? x = ? x1. This

geodesic is a limiting case of those obtained for α ?= 0, and so does not affect the conclusion above

regarding the surface formed by the null geodesics.

1,x−= γτ + δ,? x =?Aτ +?B. These geodesics

9

Page 10

where the constant C can be written in terms of the initial and final points as

C = −

Φ

4∆x+sin∆x+

(54)

Thus the distance measured along a timelike geodesic will be

?x2

x1

√−ds2=

?

−Φ

∆x+

sin∆x+= 2√−Y2√

∆x+≡

√−σ2

(55)

We observe that the argument of the Hankel function in I+is m√−σ2. Overall we

can write

I+= −π2

16Θ(∆x+)

1

(2π)6

?sin∆x+

∆x+

?−416m4

(−σ2)2H(1)

−4(m√−σ2) (56)

which differs from (52) only by the factor (sin∆x+

Let us now observe the relation to the flat space propagator in another way. In the

massless case we had found that the relation to the flat space propagator was directly

seen when using the variable Φ. Define

∆x+ )4and the replacement s2

flat→ σ2.

˜ m2= m2 ∆x+

sin∆x+

(57)

Then we observe that

I+= −π2

16Θ(∆x+)

1

(2π)6

16˜ m4

(−Φ)2H(1)

−4(˜ m√−Φ)(58)

which differs from (52) only by the replacements m → ˜ m and s2

flat→ Φ.

6Discussion

We have obtained propagators for the massless and massive scalar fields in the pp-wave

geometry obtained from AdS5× S5. This is the geometry seen by string states moving

fast around the diameter of the S5, while staying in the vicinity of the origin in global

AdS5. We have noted several relations between these propagators and the corresponding

flat space expressions. These relations should help us to guess the form of propagators

for higher spin fields. The propagators for other spaces AdSp× Sqcan be obtained by

direct inspection of our results here – the different dimension leads to a change in the

number of harmonic oscillators obtained from the transverse coordinates xi, but there is

no other essential change in the computation.

The space AdS5× S5is locally conformally flat, so for massless fields one should be

able to recover the propagator in the pp-wave from the flat space result.5(We have not

5We thank D. Berenstein for this comment.

10

Page 11

taken a conformally coupled scalar, but the curvature scalar R of the geometry vanishes

so we do not have to consider a term Rφ2in the Lagrangian for a scalar field φ.) Working

with such conformal maps may make it more difficult however to follow the behavior of

the propagator through the singularities at ∆x+= nπ, since the conformal map is local

rather than global. The massive propagator has a length scale set by the mass, and so

cannot be obtained by a conformal mapping to flat space.

The pp-wave has the property that light rays can reach the boundary and return

in a finite time, a property that is also shared by AdS spaces. We thus need to have

a boundary condition at infinity which reflects the energy of the waves back into the

geometry. The fact that we have used normalizable wavefunctions in constructing the

propagator implies that we have Dirichlet boundary conditions at infinity.

One of the interesting features of the propagators is the appearance of the function

sin∆x+, which vanishes whenever ∆x+= nπ. Let us write

x+=1

2(t + ψ), x−=1

2(t − ψ)(59)

The pp-wave geometry (1) is infinitely extended in the coordinate ψ along the wave.

However when we consider this geometry as a part of AdS5×S5then we find that ψ is an

angular variable on a circle. The point x+= π,x−= xi= 0 is at ψ = π, halfway around

the S5. The point x+= 2π,x−= xi= 0 is the same as the point x+= 0,x−= xi= 0.

Null geodesics moving on the S5cross the line xi= 0 twice in each revolution, at the

points x+= 0,x+= π.

Finally we comment on the physical problem that we would like to address with the

help of this propagator (and similar propagators for other fields). In [2] string states were

considered that move fast around the S5. If we consider higher and higher excitations of

this string, then there can come a point where the string collapses to a black hole under

its self-gravitation. Such an effect was studied for strings in flat space in [5]. Let us note

here some aspects of this problem in the pp-wave geometry.

In flat space the size of the free string is given by a ‘random walk’ made out of string

bits. In the pp-wave geometry there is a potential that drives the string to the center

xi= 0. There is no potential confining the direction ψ along the wave, so the string can

extend into a cylindrical shaped object lying along the central axis of the pp-wave.

Let us assume such a geometry for the string state, and consider the effects of self-

interaction created by a field with a propagator that behaves like that of the massless

scalar field considered here. (We expect the dilaton, 2-form gauge field an the graviton

to have propagators that are similar except for their index structure.) Note that the

massless propagator is a function of the variable Φ, and consider the propagator between

points that are near the axis xi= 0. The fact that Φ ∼ −4∆x−sin∆x+implies that the

retarded Green’s function behaves like an inverse power of sinx+instead of an inverse

power of x+.

Now consider applying such a propagator to find the self-interaction of a time inde-

pendent distribution. We find a ‘resonant interaction’ between points that are separated

11

Page 12

by ∆ψ = nπ. Such an interaction permits solutions that are independent of the coor-

dinate ψ, but it also suggests that for suitable interactions one may get lower energy

configurations that are periodic under ψ → ψ + nπ rather than independent of ψ. We

hope to return to a more detailed discussion of this problem elsewhere.

Acknowledgments

We are grateful to David Berenstein, Camillo Imbimbo, Oleg Lunin and Amit K. Sanyal

for several helpful comments. This work is supported in part by DOE grant DE-FG02-

91ER40690.

References

[1] M. Blau, J. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, Class. Quant. Grav.

19, L87 (2002) [arXiv:hep-th/0201081]; M. Blau, J. Figueroa-O’Farrill, C. Hull and

G. Papadopoulos, JHEP 0201, 047 (2002) [arXiv:hep-th/0110242].

[2] D. Berenstein, J. M. Maldacena and H. Nastase, JHEP 0204, 013 (2002)

[arXiv:hep-th/0202021].

[3] B. Allen and M. Turyn, Nucl. Phys. B 292, 813 (1987), E. D’Hoker, D. Z. Freed-

man, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B 562, 330 (1999)

[arXiv:hep-th/9902042], H. Liu and A. A. Tseytlin, Phys. Rev. D 59, 086002 (1999)

[arXiv:hep-th/9807097], A. M. Ostling “The Propagator For Graviton Modes in Su-

pergravity On ADS5XS5,” Master’s Thesis, UIUC, 1999

[4] M. Li, arXiv:hep-th/0205043.

[5] G.T.Horowitzand J.Polchinski,Phys.Rev.D

57, 2557(1998)

[arXiv:hep-th/9707170], T. Damour and G. Veneziano, Nucl. Phys. B 568, 93 (2000)

[arXiv:hep-th/9907030].

[6] I.S. Gradshteyn and I.M. Ryzhik, “Table of Integrals, Series and Products”, 4th Ed.

(1980), Academic Press, Orlando, Pg. 340, Equation 11.

[7] H.S. Bateman, “Higher Transcendental Functions”, Vol II, (1953), McGraw-Hill New

York, Pg. 194, Equation 22.

12