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arXiv:hep-th/0508199v3 15 Feb 2006

Perturbative self-interacting scalar field theory: a differential equation

approach.

R. da Rocha∗and C. H. Coimbra-Ara´ ujo†

Instituto de F´ ısica Gleb Wataghin,

Universidade Estadual de Campinas,

CP 6165, 13083-970

Campinas, SP, Brazil.

E. Capelas de Oliveira‡

Departamento de Matem´ atica Aplicada, IMECC,

Unicamp, CP6065, 13083-859, Campinas, SP, Brazil.

We investigate the partition function related to a φ4-scalar field theory on a n-dimensional

Minkowski spacetime, which is shown to be a self-interacting scalar field theory at least in 4-

dimensional Minkowski spacetime. After revisiting the analytical calculation of the perturbative

expansion coefficients and also the approximate values for suitable limits using Stirling’s formulæ,

we investigate a spherically symmetric scalar field in a n-dimensional Minkowski spacetime. For the

first perturbative expansion coefficient it is shown how it can be derived a modified Bessel equation

(MBE), which solutions are investigated in one, four, and eleven-dimensional Minkowski spacetime.

The solutions of MBE are the first expansion coefficient of the series associated with the partition

function of φ4-scalar field theory. All results are shown graphically.

I.INTRODUCTION

The results presented in this article are based on Witten’s proposed questions, solved in [2], by P. Deligne,

D. Freed, L. Jeffrey, and S. Wu, concerning perturbative φ4-scalar field theory (PSFT). We extend their

solutions and show that the first perturbative coefficient can be obtained from modified Bessel functions of

first kind. This article is organized as follows: In Sec. (II) the partition functional is exhibited in the context

of a self-interacting φ4-scalar field theory and a perturbative solution for the partition function is obtained,

in the light of the solutions obtained by Deligne, Freed, Jeffrey, and Wu [2]. In Sec. (III) the method

presented is shown to hold for a spherically symmetric scalar field in n-dimensional Minkowski spacetime.

The perturbative expansion coefficients are derived, and the first coefficient is shown to constrain the scalar

field in a MBE, which solutions are used to obtain the first perturbative expansion coefficient associated

with the partition function of PSFT, for the particular cases of 1-, 4-, and 11-dimensional spacetime. Nu-

merical integration permits us to accomplish the expansion coefficient for 2- and 4-dimensional spacetime.

In Appendix we plot the main results.

II. PERTURBATIVE φ4-SCALAR FIELD THEORY

We begin with the general form of the scalar Lagrangian density

L =1

2∂µφ∂µφ − V (φ), (1)

∗Electronic address: roldao@ifi.unicamp.br

†Electronic address: carlosc@ifi.unicamp.br

‡Electronic address: capelas@ime.unicamp.br

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where φ = φ(x) is a Hermitian scalar field and V = V (φ(x)) denotes the scalar potential. For instance, in a

self-interacting 4-dimensional scalar field theory it is well-known that [1]

V (φ) =1

2m2φ2+1

4!λφ4,(2)

where λ is the self-coupling constant and m2denotes the mass parameter. Euler-Lagrange equation of motion

gives, from eq.(1),

∂µ∂µφ = −∂V (φ)

∂φ

.(3)

The partition function associated with eqs.(1,2), can be written as

Z(λ) =

?∞

−∞

exp

?

−1

2φ2−1

4!λφ4

?

dφ.(4)

As we want to investigate the perturbative character of the scalar field theory given by the Lagrangian

density in eq.(1) with scalar potential given by eq.(2), it is now possible to expand the partition function

Z(λ) in terms of a series as

Z(λ) =

∞

?

k=0

ckλk,ck∈ R.(5)

Using Taylor’s formulæ it can be shown that [2]

ck=(−1)k

(4!)kk!

?∞

−∞

exp

?

−1

2φ2−1

4!φ4k

?

dφ.(6)

Now define

f(A) =

?∞

?∞

√2π exp

−∞

exp

?

?

?1

−1

−1

2φ2+ Aφ

?

?

dφ

=

−∞

exp

2(φ + A)2

?

exp

?1

2A2

?

dφ

=

2A2

.(7)

It is clear that the integral involving the coefficient ckin eq.(6) can be written as

∂(4k)f(A)

∂A(4k)

|A=0=

√2π

(4k)!

(2k!)4k,(8)

from which it follows that [2]

ck=(−1)k(4k)!√2π

k!(24)k(2k)!4k= (−1)k

√2π

(24)k(4k − 1)!!(9)

Then, perturbatively for large k the expansion in eq.(5) gives for the coefficient ckthe expression

ck≈ (−1)k√2k−1/2

?2k

3

?k

e−k

(10)

where Stirling’s formulæ n! ≈√2πnnne−nis used [2, 5].

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III.

n-DIMENSIONAL SPHERICALLY SYMMETRIC PATH INTEGRAL: PERTURBATIVE

METHOD

Considering a spherically symmetric scalar field φ = φ(r) in a n-dimensional spacetime, where r denotes

the radial coordinate, the Lagrangian density is given by

L =1

2?dφ?2+1

2m2φ2+1

4!α4−nλφ4, (11)

where m denotes the mass associated with the scalar field and α is an arbitrary mass constant parameter

introduced in order to leave the self-coupling constant α dimensioless in n-dimensions. The term dφ denotes

the differential operator acting on the scalar field φ as dφ =∂φ

The standard path integral representation of a theory defined by an action S =?L is obtained by defining

?

where the?Dφ expression is just a shorthand for the product

?

x

∂rdr.

the quantity Z, the partition function, as

Z =exp(iS)Dφ(12)

Dφ =

?

?

dφ(x). (13)

This product is taken over all infinite spacetime points x in the volume of the system being described. We

immediately spot similarities between the path integral in eq.(12) and the partition function of statistical

mechanics Z = tr[exp(−S)] [4]. The above integral over all field configurations is precisely analogous to a

trace over all degrees of freedom of a system. However, the imaginary exponential factor in the path integral

does not lend itself to a probabilistic interpretation in the same way that with the partition function. A

standard trick used for formulating theories on a lattice is employed, which is to move into imaginary or

Euclidean time [4]. The Euclidean spacetime metric tensor is given by gij = δij, where δij denotes the

Kronecker tensor. The path integral is written as

Z =

?

exp(−S)Dφ(14)

where

S =

? ?1

2∂µφ∂µφ +1

2m2φ2+1

4!α4−nλφ4

?

dt ∧ dx1∧ ··· ∧ dxn−1

(15)

with ∂µ∂µ=

n−1

?

i=1

∂2

∂x2

i

+∂2

∂t2. The exponential function in the path integral is now free of imaginary factors and

can be interpreted as a probability distribution, thereby making the connection with the standard partition

function from statistical mechanics. The Euclidean formulation is crucial for the operation of lattice Monte

Carlo investigations [3]. Hereon

dη = dt ∧ dx1∧ ··· ∧ dxn−1

(16)

denotes the n-dimensional volume element in a given local coordinate chart.

Proceeding as in Sec. (II), by expanding the partition function as in eq.(5) it follows that

ck =

(−1)k

(4!)kk!

??

?

exp

?

−1

−1

2

?

(?dφ?2+ m2φ2)dη

???

αn−4φ4dη

?k?

Dφ (17)

=

(−1)k

(4!)kk!

exp

?

2

?

(?dφ?2+ m2φ2)dη + kln

??

αn−4φ4dη

??

Dφ.(18)

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In order to find the exponent critical points in eq.(18), which is equivalent to have knowledge of the large k

behavior of the coeffiecients ck[2], one takes the derivative with respect to the r coordinate in both terms

of the exponent −1

2

?

(?dφ2? + m2φ2)dη + kln

??

αn−4φ4dη

?

, yielding:

??−∆φ − m2φ?dφ

drdη +4k?φ3dφ

drdη

?φ4dη

(19)

where ∆ denotes the Laplacian operator. At the critical points of the exponent, the integrand in eq.(19)

equals zero, and then

(∆ + m2)φ =

4kφ3

?φ4dη.

(20)

As φ = φ(r) is a radial scalar field, the Laplacian is given by

∆φ = −

1

rn−1

d

dr

?

rn−1dφ

dr

?

, (21)

and eq.(20) can be written as

d2φ

dr2+n − 1

r

dφ

dr− m2φ +

kφ3

π2?∞

0r3φ4(r)dr= 0 (22)

which can be led to

d2φ

dr2+n − 1

r

dφ

dr− m2φ + kφ3= 0(23)

if the rescaling φ ?→?π2?∞

lim

[2]. Denoting hereon

0r3φ4(r)dr?−2φ is imposed [2]. This rescaling must be finite, and so the condition

r→0

r→∞φ(r) = 0 must holds. Besides, lim

dφ

dr= 0 in order that the derivative dφ/dr to make sense in r = 0

V (φ) =k

4φ4−m2

2φ2,(24)

for n = 1 eq.(23) is Newton’s second law associated with the potential V (φ) given by eq.(24):

d2φ

dr2+dV (φ(r))

dφ

= 0, (25)

For n > 1 we have

d2φ

dr2+dV (φ(r))

dφ

+n − 1

r

dφ

dr= 0. (26)

When k = 0, eq.(23) can be written as

d2φ

dr2+n − 1

r

dφ

dr− m2φ = 0 (27)

which can be exactly solved if we first multiply eq.(28) by r2, yielding

r2d2φ

dr2+ r(n − 1)dφ

dr− m2r2φ = 0.(28)

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Now, let us consider φ(r) = rβξ(r), and on substituting this ansatz in the equation above, which seems to

be the best adapted to our investigation, it follows that

r2d2ξ(r)

dr2

+ (2β + n − 1)rdξ(r)

dr

+ [β(β − 1) + (n − 1)β]ξ(r) − m2r2ξ(r) = 0. (29)

We now are left with the task of determining a suitable choice of the parameter β in eq.(29) in such a way

that the coefficient of the term rdξ(r)

dr

With this choice, it can be immediately shown that β = 1 −n

to be 1, i.e., we choose β in order that eq.(29) can be led to a MBE.

2. Eq.(29) is then expressed as

r2d2ξ(r)

dr2

+ rdξ(r)

dr

−

??

1 −n

2

?2

+ m2r2

?

ξ(r) = 0.(30)

This is the MBE, which has as solutions the modified Bessel functions of the first kind I1−n

second kind K1−n

2(mr) and of the

2(mr) [6]. Finally the self-interacting radial scalar field is given, for k = 0, by

φ(r) = r1−n

2I1−n

2(mr).(31)

It can be shown that the solution above is regular at the origin, at least for n even [5]. The graphics in

Appendix show the behavior of such function for each n. We use unitary field mass, and assume m = 1

without loss of generality.

From eq.(6) it follows that the first partition function perturbative expansion coefficient c0 in eq.(5) is

given by

c0= lim

ǫ→0

?∞

ǫ

exp

?

−1

2r2−nI2

1−n

2(mr)

?dI1−n

2

dr

dr.(32)

For n = 1 we have

c0=

?

2

πmlim

ǫ→0

?∞

ǫ

1

rexp

?

−

?

r

2πmsinh(mr)

? ?

−sinh(mr)

2r

+ mcosh(mr)

?

dr(33)

For n = 4, consisting of the usual Minkowski spacetime R1,3, it follows that c0is given by

c0=

?∞

0

exp

−m3

16

∞

?

j=0

(mr)2j

4jj!(2 + j)!

2 ?∞

?

p=0

(2p + 1)(mr)2p

4pp!(2 + p)!

?

dr,(34)

and numerical integration gives us

c0= 3.85378.(35)

For n = 2, it follows from eq.(32) that

c0= 0.39769.(36)

Finally, for n = 11, where now the scalar field is a 11-dimensional Minkowski spacetime-valued function,

it follows that

c0 =

?

×{sinh(mr)(−14m4r5− 9m4r4− 30m2r3− 405m2r2− 945) + mrcosh(mr)(2m4r5+ 30m2r3+ 90m2r2+ 945)}

This formulæ can be useful in investigating the KK gravitational waves propagation modes in brane-world

scenario [7, 9].

1

2πm9lim

ǫ→0

?∞

ǫ

dr

rexp

??

2

π(mr)9(m4r4+ 45m3r2+ 105)sinh(mr) − mr(10m2r2+ 105)cosh(mr)

?