Sep 22, 2011 - 3 (2006) 149. [7] M. R. Douglas, N. A. Nekrasov, Rev. Mod. Phys. 73(2001) 977. [8] X. Calmet, A. Kobakhidze, Phys. Rev. D 72 (2005) 045010.
Path Integral Quantization of Noncommutative Complex Scalar Field Farid Khelili 20 Aout 55 Skikda University, Skikda, Algeria
arXiv:1109.4741v1 [hep-th] 22 Sep 2011
Abstract Using noncommutative deformed canonical commutation relations, a model describing a noncommutative complex scalar field theory is considered. Using the path integral formalism, the noncommutative free and exact propagators are calculated to one-loop order and to the second order in the parameter of noncommutativity. Dimensional regularization was used to remove ultraviolet divergences that arise from loop graphs. It has been shown that these divergences may also be absorbed into a redefinition of the parameters of the theory. PACS numbers: Keywords: Noncommutative space, Noncommutative scalar field, Path integral, Propagators, Renormalization, Dimensional regularization.
1
1.
INTRODUCTION
Noncommutative geometry is presently one of the most important and extremely active area of research in theoretical physics, there is now a common belief that the usual picture of space-time as a smooth pseudo-Riemannian manifold should breakdown at very short distances of the order of the Planck length, due to the quantum gravity effects. The concept of noncommutative space-time was suggested very early on by the founding fathers of quantum mechanics and quantum field theory. This was motivated by the need to remove the divergences which had plagued quantum electrodynamics. However, this suggestion was ignored [1] . In recent years, the idea of noncommutative space-time has attracted considerable interest, and has penetrated into various fields in physics and mathematical physics, it leads to the investigation of some new and more fundamental physical and mathematical notions, the motivation for this kind of investigation is that the effects of noncommutativity of space may appear at very short distances of the order of the Planck length, or at very high energies, this may shed a light on the real microscopic geometry and structure of our universe [1]-[14]. One of the new features of noncommutative field theories is the UV/IR mixing phenomenon, in which the physics at high energies affects the physics at low energies, which does not occur in quantum field theories in which the coordinates commute [1] [12] [13]. Thus in the noncommutative space-time approach, the dynamical variables become operators, and, therefore, the formalism of the quantum field theory constructions must be modified. The discovery of noncommutative geometry has allowed the exploration of new directions in theoretical physics, in particular, several aspects of noncommutative quantum mechanics and two-dimensional noncommutative harmonic oscillators are an extremely active area of research and have been discussed extensively from different points of view [15]-[34]. Our paper is organized as follows: In Section 2, we consider a noncommutative action for a complex scalar field with self interaction, in section 3, we briefly recall basic results of path integral derivation of propagator and renormalization of ϕ4 − theory, in section 4, we consider the path integral derivation of the noncommutative free and exact propagators, in section 5 we consider the dimensional regularization of the noncommutative exact propagator, Finally in section 6, we draw our conclusions.
2
2.
NONCOMMUTATIVE ACTION
In [35] a model describing a noncommutative complex scalar field theory, based on noncommutative deformed canonical commutation relations, was considered, the model has been quantized via the Peierls bracket [37]. Here we will use the path integral formalism to quantize the noncommutative complex scalar field theory. Consider a complex scalar field Φ (x) with Lagrangian density given by [36]-[40] L = − (∂µ Φ)∗ (∂ µ Φ) − m2 Φ∗ Φ − g (Φ∗ Φ)2
(1)
where m is the mass of the charged particles, and g is a positive parameter. The metric signature will be assumed to be − + +..., in what follows, we take ~ = c = 1. The complex scalar field can be quantized using the canonical quantization rules, for this we express it in terms of its real and imaginary parts as Φ =
√1 2
(ϕ1 + iϕ2 ), where ϕ1 , ϕ2 are
real scalar fields; in terms of these real scalar fields the Lagrangian density reads 1 1 1 1 1 L = − (∂µ ϕa )2 − m2 (ϕa )2 − g (ϕa ϕa )2 = − (∂µ ϕa )2 − µ2 [ϕ] (ϕa )2 2 2 4 2 2
(2)
where µ2 [ϕ] = m2 + 12 g (ϕa )2 . Let πa be the canonical conjugate to ϕa πa =
∂L · = ϕa . ∂ ϕa
(3)
The Hamiltonian density reads then → 2 1 2 1 1 − 2 H = πa ϕa − L= (πa ) + ∇ϕa + µ [ϕ] (ϕa )2 2 2 2 ·
(4)
In the canonical quantization the canonical variables ϕa and the canonical conjugates πa are assumed to be operators satisfying the canonical commutation relations
→ → → → [ϕa (t, − x ) , πb (t, − y )] = iδab δ 3 (− x −− y)
(5)
→ → [ϕa (t, − x ) , ϕb (t, − y )] = 0 → → [πa (t, − x ) , πb (t, − y )] = 0 It is well known, since the birth of quantum field theory in the papers of Born, Dirac, Fermi, Heisenberg, Jordan, and Pauli, that the free field behaves like an infinite number 3
of coupled harmonic oscillators [36], using this analogy between free fields and an infinite number of coupled harmonic oscillators, one can impose non commutativity on the configuration space of dynamical fields ϕa , to do this we recall that the two-dimensional harmonic oscillator noncommutative configuration space can be realized as a space where the coordinates x ba , and the corresponding noncommutative momentum pba , are operators satisfying
the commutation relations
[b xa , x bb ] = iθ2 εab
[b pa , pbb ] = 0
[b xa , pbb ] = iδab
(6)
where θ is a parameter with dimension of length, and εab is an antisymmetric constant matrix. It is well known that this noncommutative algebra can be mapped to the commutative Heisenberg-Weyl algebra [32]-[34] [xa , xb ] = 0
[pa , pb ] = 0
[xa , pb ] = iδab
(7)
through the relations 1 x ba = xa − θ2 εab pb 2
pba = pa
(8)
To impose non commutativity on the configuration space of dynamical fields ϕa , we assume that the noncommutative canonical variables ϕ ba and the noncommutative canonical
conjugates π ba satisfy the noncommutative commutation relations → → → → [ϕ ba (t, − x ),π bb (t, − y )] = iδ 3 (− x −− y ) δab
(9)
→ → → → [ϕ ba (t, − x),ϕ bb (t, − y )] = iθεab δ 3 (− x −− y) → → [b πa (t, − x ),π bb (t, − y )] = 0
where θ is the parameter of noncommutativity, which is assumed to be a constant, and εab is a 2 × 2 real antisymmetric matrix ε12 = −ε21 = 1 The noncommutative Hamiltonian density is assumed to have the form 4
(10)
→ 2 1 2 1 − b 1 (b ∇ϕ ba + µ [ϕ] πa )2 + b (ϕ ba )2 H= 2 2 2
(11)
By generalizing the noncommutative harmonic oscillator construction an extension of quantum field theory based on the concept of noncommutative fields satisfying the noncommutative commutation relations (9) has been proposed in [27]-[29], where the properties and phenomenological implications of the noncommutative field has been studied and applied to different problem including scalar, gauge and fermionic fields [27]-[31]. Our approach is different, it is based on the relation between the noncommutative variables ϕ ba and π ba and the canonical variables ϕa and πa .
It is easy to see that the noncommutative commutation relations (9) can be mapped
to the canonical commutation relations (5) if the noncommutative variables ϕ ba and π ba are
related to the canonical variables ϕa and πa by the relations
1 ϕ ba = ϕa − θεab πb 2
(12)
π ba = πa
Using these transformations, the noncommutative Hamiltonian density eq(11) can be rewritten, up to a total derivative term and up to second order in the parameter θ, as
where
b 1 π ∼ Mπ − 1 θ2 π ∼ Dπ + θπ ∼ Nϕ+ 1 ϕ∼ Bϕ + O θ3 H= 2 8 2
1 σε = M∼ M= I+ θ2 m2 I − gεb 4 δ2 1 ∼ 2 σ bab = (ϕ ϕ) = ϕ∼ ϕδab + 2ϕa ϕb δϕa δϕb 4 σ b = (ϕ∼ ϕ) I + 2M [ϕ] , Mab [ϕ] = ϕa ϕb → − D= ∇ 2 I = D∼ → − 1 N= m2 − ∇ 2 + g (ϕa )2 ε = −N∼ 2 →2 1 − 2 2 B= m − ∇ + g (ϕa ) I = B∼ 2 5
(13)
(14)
with I denotes the 2 × 2 unit matrix, and A∼ denotes the transpose of the operator A. From now on we keep only the modifications due to the noncommutativity up to second order in the parameter θ. ·
The relation between πa and ϕa is given by ·
ϕa (x) = b = where H
R
b δH δπa (x)
(15)
b Using the expression of H b and the symmetry properties of the d3 xH.
operators M = M∼ and D = D∼ , one gets
1 · ϕa (x) = Mab πb (x) − θ2 Dab πb (x) + θNab ϕb (x) 4
(16)
From this relation we get the following iterative expression of πa · 1 πa = Kab ϕb − θNbc ϕc + θ2 Kab Dbc πc 4
(17)
where K is the inverse of the matrix M
1 σε K = M−1 = I− θ2 m2 I − gεb 4
(18)
Using the expression of the matrix K one gets, by iteration, the following expression of πa
1 2 · πa = I+ θ D ϕb − θNab ϕb 4 ab → − ∼ where D = D− (m2 I − εb σε) = ∇ 2 I − (m2 I − gεb σε) = D .
(19)
We note that the noncommutative Hamiltonian density can be derived from the following noncommutative Lagrangian density 1 1 2 1 ·∼ · · b L = ϕ I+ θ D ϕ + θϕ∼ Nϕ − ϕ∼ B+θ2 N2 ϕ 2 4 2
(20)
b = πa ϕ· a − H. b To get this expression we have via the usual Legendre transformation L
used the symmetry properties of the operators D, N and B.
6
3.
PROPAGATOR AND RENORMALIZATION
In this section we briefly recall basic results of path integral derivation of propagators and renormalization of ϕ4 − theory [40] [39] [36] [41]-[44]. Let us consider the theory of a real self-interacting scalar field specified by an action of the form S [ϕ] =
Z
1 1 1 d x − [∂µ ϕ (x)]2 − m2 [ϕ (x)]2 − g [ϕ (x)]4 2 2 4! 4
The action can be written as
S [ϕ] = S (0) [ϕ] + S (1) [ϕ]
(21)
(22)
where S (0) [ϕ] is the free action S
(0)
1 [ϕ] = − 2
Z
d4 xd4 yϕ (x) D (x, y) ϕ (y)
(23)
and S (1) [ϕ] is the interaction term S
(1)
1 [ϕ] = − g 4!
Z
d4 x [ϕ (x)]4
(24)
with D (x, y) = m − δ 4 (x, y) = 2
Z
iq(x−y) d4 q 2 2 m + q e = D (y, x) (2π)4
(25)
The free propagator ∆ (x, y) is defined as the vacuum expectation value [36] [41]-[44] − i∆ (x, y) = h0| T {Φ (x) Φ (y)} |0i
(26)
where h0| T {A [Φ]} |0i denotes the vacuum expectation value of the chronological product T {A [Φ]}, Φ (x) denotes the free quantum field operator corresponding to ϕ (x), and T is the time-ordering operator. The vacuum expectation value eq(26) can be expressed as a Feynman functional integral [36] [41] [42]
− i∆ (x, y) = h0| T {Φ (x) Φ (y)} |0i =
RY (0) dϕ (x) [ϕ (x) ϕ (y)] eiS [ϕ] x
RY x
7
dϕ (x) eiS (0) [ϕ]
(27)
The functional integrals in eq(27) can now be performed with the help of the identity [36] − 12 Z Y iD iS (0) [ϕ] dϕ (x) [ϕ (xl1 ) ϕ (xl2 ) ...ϕ (x2N )] e = det 2π x
X
Y
pairings pairs(lj ,lk ) of (l1 ,l2 ,...,l2N )
−iD −1 xlj , xlk (28)
where the sum is over all ways of pairing the indices l1 , l2 , ..., l2N , with two pairings being considered the same if they differ only by the order of the pairs, or by the order of indices within a pair, and D −1 (x, y) is the inverse of the matrix D (x, y). Direct calculations give the following expression for the propagator − i∆ (x, y) = −iD −1 (x, y)
(29)
to get the inverse of the matrix D (x, y) we write D (x, y) as a Fourier integral D (x, y) =
Z
d4 q iq(x−y) 4 D (q) e (2π)
(30)
where D (q) = m2 + q 2 − iǫ
(31)
Eq (29) can be rewritten as an integral equation Z
d4 yD (x, y) ∆ (y, z) = δ (x, z)
(32)
The solution of eq (32) is then ∆ (x, y) = where ∆ (q) = D −1 (q) =
1 m2 +q 2 −iǫ
Z
d4 q iq(x−y) 4 ∆ (q) e (2π)
(33)
is the free-field propagator, and the iǫ terms have the
effect of making the inverse well-defined for all real values of q. The exact propagator ∆′ (x, y) is given by [36] [41] [42]
− i∆′ (x, y) = h0| T {Φ (x) Φ (y)} |0i =
RY dϕ (x) [ϕ (x) ϕ (y)] eiS[ϕ] x
RY dϕ (x) eiS[ϕ]
(34)
x
In order to evaluate the functional integral eq(34), we make a Taylor expansion in powers of g 8
eiS[ϕ] = eiS
(0) [ϕ]+iS (1) [ϕ]
= eiS
(0) [ϕ]
1 + iS (1) [ϕ] + ...
(35)
Using this Taylor expansion in eq(34) we obtain the following expressions Z Y Z Y (0) iS[ϕ] dϕ (x) [ϕ (x) ϕ (y)] e = dϕ (x) [ϕ (x) ϕ (y)] eiS [ϕ] x
x
i − g 4!
Z
and Z Y x
iS[ϕ]
dϕ (x) e
=
Z Y
(36)
Z Y (0) dz dϕ (x) ϕ (x) ϕ (y) [ϕ (z)]4 eiS [ϕ] 4
x
iS (0) [ϕ]
dϕ (x) e
x
i − g 4!
Z
Z Y (0) dz dϕ (x) [ϕ (z)]4 eiS [ϕ] 4
(37)
x
The functional integrals in eq(36) and eq(37) can now be performed with the help of the identity eq(28), the exact propagator ∆′ (x, y) can be written as ′
∆ (x, y) = where
Z
d4 q ′ ∆ (q) eiq(x−y) (2π)4
∆′ (q) = ∆ (q) + ∆ (q) Π∗Loop (q) ∆ (q)
(38)
(39)
and Π∗Loop (q) is given by the divergent integral Π∗Loop
i (q) = g 2
Z
1 d4 p 4 (2π) m2 + p2 − iǫ
(40)
such ultraviolet divergences are typical of loop graphs. To deal with these divergent integrals that appear in quantum field theory, one can use some sort of regularization technique that makes these integrals finite. Dimensional regularization is the most convenient method for regulating divergent integrals, the idea is to treat the loop integrals as integrals over D-dimensional momenta, and then take the limit D → 4, it turns out that the singularity of 1-loop graphs are simple poles in D = 4 [40] [39] [36] [45] [41]-[44]. First we generalize the 4-dimensional action eq(28) to D-dimensions [36] [42] (0)
(1)
S [ϕB ] = SB [ϕB ] + SB [ϕB ] 9
(41)
with
(0) SB
and
1 [ϕB ] = − 2
Z
dD xdD yϕB (x) D (B) (x, y) ϕB (y) Z iq(x−y) 4 dD q 2 2 (B) 2 m + q e D (x, y) = mB − δ (x, y) = B (2π)D (1) SB
1 [ϕB ] = − µ4−D gB 4!
Z
dD x [ϕB (x)]4
(42) (43)
(44)
where the scalar field ϕB is a bare field, mB is the bare mass, and µ is an arbitrary mass parameter introduced to keep the bare coupling constant gB dimensionless. The bare quantities, such as ϕB , mB and gB , are objects which are useful in the intermediate steps of the calculations, but they have no physical meaning. They are just chosen so that they cancel the divergences and leave us with the desired renormalized quantity [39] [36] [43]. The physical or renormalized scalar field ϕ, mass m and coupling constant g are defined by [39] [36] [43] [42]
1
ϕ=Z − 2 ϕB −1
g = Zg 2 g B
, m2 = m2B + δm2 , Zg =
(1 + B) Z2
(45) (46)
with Z and δm2 to be chosen so that the propagators of the renormalized fields have poles in the same position and with the same residues as the propagators of the free fields in the absence of interactions. The action may then be written in terms of renormalized quantities, as S [ϕ] = S (0) [ϕ] + S (1) [ϕ] + S (c) [ϕ]
(47)
where Z 1 S [ϕ] = − dD xdD yϕ (x) D (x, y) ϕ (y) 2 Z 1 4−D (1) g dD x [ϕ (x)]4 S [ϕ] = − µ 4! Z iq(x−y) 4 d4 q 2 2 2 e D (x, y) = m − δ (x, y) = 4 m +q (2π) (0)
10
(48) (49) (50)
and S
(c)
1 [ϕ] = − (Z − 1) 2
Z
1 d xd yϕ (x) D (x, y) ϕ (y)+ Zδm2 2 D
D
Z
Z B 4−D d xϕ (x) ϕ (x)− µ g dD x [ϕ (x)]4 4! (51) D
The renormalized exact propagator ∆(R) (x, y) is given by
− i∆(R) (x, y) = h0| T {Φ (x) Φ (y)} |0i =
RY x
dϕ (x) [ϕ (x) ϕ (y)] eiS[ϕ] RY dϕ (x) eiS[ϕ]
(52)
x
We would like to evaluate the path integral for this theory, we now make a Taylor expansion in powers of g eiS[ϕ] = eiS
(0) [ϕ]+iS (1) [ϕ]
= eiS
(0) [ϕ]
1 + iS (1) [ϕ] + iS (c) [ϕ] + ...
(53)
Using this Taylor expansion in eq(52) we obtain the following expressions Z Y Z Y (0) iS[ϕ] dϕ (x) [ϕ (x) ϕ (y)] e = dϕ (x) [ϕ (x) ϕ (y)] eiS [ϕ] x
and
x Z Z Y (0) i 4−D g dD z dϕ (x) ϕ (x) ϕ (y) [ϕ (z)]4 eiS [ϕ] − µ 4! x Z Z Y i (0) D D − (Z − 1) d zd τ D (z, τ ) dϕ (x) [ϕ (x) ϕ (y) ϕ (z) ϕ (τ )] eiS [ϕ] 2 x Z Z Y (0) i dϕ (x) ϕ (x) ϕ (y) [ϕ (z)]2 eiS [ϕ] + Zδm2 dD z 2 x Z Z Y (0) iB 4−D D µ g d z dϕ (x) ϕ (x) ϕ (y) [ϕ (z)]4 eiS [ϕ] − 4! x
Z Y Z Z Y Z Y i 4−D (0) D iS[ϕ] iS (0) [ϕ] g d z dϕ (x) e = dϕ (x) [ϕ (z)]4 eiS [ϕ] dϕ (x) e − µ 4! x x x Z Z Y i (0) − (Z − 1) dD zdD τ D (z, τ ) dϕ (x) [ϕ (z) ϕ (τ )] eiS [ϕ] 2 Z Z Y x 1 (0) dϕ (x) [ϕ (z)]2 eiS [ϕ] + Zδm2 dD z 2 x Z Z Y B 4−D (0) D − µ g d z dϕ (x) [ϕ (z)]4 eiS [ϕ] 4! x 11
The functional integrals in eq(52) can now be performed with the help of the identity eq(28), the renormalized exact propagator ∆(R) (x, y) can be written as (R)
∆
(x, y) =
Z
dD q (2π)
D
∆(R) (q) eiq(x−y)
(54)
with ∆(R) (q) = ∆ (q) + ∆ (q) Π∗ q 2 ∆ (q) =
where Π∗ (q 2 ) is the self-energy function
and
1 2 + O g q 2 + m2 − Π∗ (q 2 ) − iǫ
Π∗ q 2 = − (Z − 1) m2 + q 2 + Zδm2 + Π∗Loop q 2
Π∗Loop
q
2
i = µ4−D g 2
Z
dD p 1 4 2 (2π) p + m2 − iǫ
(55)
(56)
(57)
The condition that m2 is the true mass of the particle is that the pole of the propagator should be at q 2 = −m2 , so that [36] Π∗ −m2 = 0
(58)
Also, the condition that the pole of the propagator at q 2 = −m2 should have a unit residue (like the uncorrected propagator) is that d ∗ 2 Π q =0 dq 2 q 2 =−m2
(59)
These conditions allow us to evaluate Z and δm2 [36] Zδm2 = −Π∗Loop −m2
and
d ∗ ΠLoop q 2 Z =1+ 2 dq
(61) q 2 =−m2
Direct calculations give the following expression for the loop integral [41]-[44] [45]
12
(60)
Π∗Loop
q
2
i = µ4−D g 2
Z
dD p m2 1 = −g 32π 2 (2π)4 p2 + m2 − iǫ
4πµ2 m2
2− D2
D Γ 1− 2
(62)
The divergence of this integral manifests itself in the pole of the Gamma function at the physical dimension D = 4. In the neighborhood of this dimension we set D = 4 − 2s
(63)
then Π∗Loop
q
2
m2 = −g 32π 2
using the relations [46] [38] [42]
Γ (s − 1) = − and
we get Π∗Loop
q
hence
2
m2 4πµ2
−s
m2 = 1 − s ln 4πµ2
2
Zδm = δm =
−Π∗Loop
Γ (s − 1)
(64)
(65)
+ O (s)
(66)
2 m2 1 m =g + 1 − γ − ln + O (s) 2 32π s 4πµ2
d ∗ 2 Z =1+ Π q Loop dq 2
2
−s
1 − γ + 1 + O (s) s
and
m2 4πµ2
2
−m
= 1 + O g2
q 2 =−m2
2 m2 m m2 1 +g 1 − γ − ln + O g2 =g 2 2 2 32π s 32π 4πµ
(67)
(68)
(69)
to this order Π∗ (q 2 ) is equal to zero
Π∗ q 2 = 0 + O g 2 13
(70)
4.
PATH INTEGRAL DERIVATION OF NONCOMMUTATIVE PROPAGATOR 4.1.
Noncommutative Free Propagator
The noncommutative action eq(20) can be rewritten as
with
Sb [ϕ] =
Z
b = Sb(0) [ϕ] + Sb(1) [ϕ] + Sb(2) [ϕ] + Sb(3) [ϕ] + Sb(4) [ϕ] + Sb(5) [ϕ] d4 xL
Z 1 eab (x, y) ϕb (y) S [ϕ] = − d4 xd4 yϕa (x) D 2 Z 1 (1) b S [ϕ] = − g d4 x [ϕa (x) ϕa (x) ϕb (x) ϕb (x)] 4 Z 1 (1) Sb(2) [ϕ] = gθ d4 xd4 yKab (x, y) [ϕa (x) ϕb (y) ϕc (x) ϕc (x)] 2 Z 1 2 (3) d4 xd4 yd4τ K(2) (x; y, τ ) [ϕa (x) ϕa (y) ϕc (τ ) ϕc (τ )] Sb [ϕ] = gθ 4 Z 1 2 (4) b d4 xd4 yd4τ K(3) (x; y, τ ) [ϕa (y) ϕa (τ ) ϕc (x) ϕc (x)] S [ϕ] = − gθ 8 Z 1 2 (5) b d4 xd4 yd4τ K(3) (x; y, τ ) εad εbc [ϕa (y) ϕd (x) ϕb (τ ) ϕc (x)] S [ϕ] = − gθ 4 b(0)
(71)
(72) (73) (74) (75) (76) (77)
eab (x, y) is the symmetric matrix where D
bba (y, x) eab (x, y) = Dab (x, y) − 1 θ2 Mab (x, y) − θNab (x, y) = D (78) D 4 Z iq(x−y) d4 q 2 2 e = Dba (y, x) (79) Dab (x, y) = δab m2 − δ 4 (x, y) = δab 4 m +q (2π) Z 2 − iq(x−y) 2 − →2 4 d4 q → 2 2 2 2 m + q m + q e Mab (x, y) = δab m − m − ∇ δ (x, y) = δab (2π)4 (80)
Mab (x, y) = Mba (y, x) (81) Z → − d4 q →2 [iq ] eiq(x−y) = N (y, x) − 2 Nab (x, y) = m2 − ∇ 2 εab ∂t δ 4 (x, y) = εab 0 ba 4 m + q (2π) (82)
(1)
while Kab (x, y) , K(2) (x; y, τ ) and K(3) (x; y, τ ) are given by 14
Z
d4 q (1) iq(x−y) = Kba (y, x) (83) 4 [εab iq0 ] e (2π) − − 1 − →2 4 − → 4 → 4 →2 4 (2) 4 2 4 K (x; y, τ ) = δ (x, τ ) m − ∇ δ (x, y) − ∇δ (x, τ ) ∇δ (x, y) − ∇ δ (x, τ ) δ (x, y) 2 (1) Kab
4
(x, y) = εab ∂t δ (x, y) =
K(2) (x; y, τ ) =
Z
1 →2 iq(x−τ ) ip(x−y) 1 → d4 q d4 p 2 → 2 (− p +− q) + − e p e 4 4 m + 2 2 (2π) (2π)
K(3) (x; y, τ ) = ∂t δ 4 (x, y) ∂t δ 4 (x, τ ) Z d4 q d4 p (3) iq(x−y) ip(x−τ ) K (x; y, τ ) = − e = K(3) (x; τ, y) 4 4 q0 p0 e (2π) (2π)
(84)
(85)
(86) (87)
e ab (x, y) is defined as the vacuum expectation The free noncommutative propagator ∆
value
e ab (x, y) = h0| T {Φa (x) Φb (y)} |0i − i∆
(88)
where h0| T {A [Φ]} |0i denotes the vacuum expectation value of the chronological product T {A [Φ]}, Φ (x) denotes the free quantum field operator corresponding to ϕ (x), and T is the time-ordering operator. The vacuum expectation value eq(26) can be expressed as a Feynman functional integral RY b(0) dϕa (x) [ϕa (x) ϕb (y)] eiS [ϕ] e ab (x, y) = h0| T {Φa (x) Φb (y)} |0i = − i∆
a,x
RY dϕa (x) eiSb(0) [ϕ]
(89)
a,x
The functional integrals in eq(27) can now be performed with the help of the identity Z Y i (0) h b dϕ (x) ϕl1 (x1 ) ϕl2 (x2 ) ...ϕl2N (x2N ) eiS [ϕ] x
"
= det
e iD 2π
!#− 12
X
Y
pairings pairs(xj lj ,xk lk ) of (x1 l1 ,x2 l2 ,...,x2N l2N )
(90) h i e −1 (xj , xk ) −iD lj ,lk
where the sum is over all ways of pairing the indices (x1 l1 , x2 l2 , ..., x2N l2N ), with two pairings being considered the same if they differ only by the order of the pairs, or by the e −1 (x, y) is the inverse of the matrix D eab (x, y) . order of indices within a pair, and D ab
15
Direct calculations give the following expression for the propagator e ab (x, y) = −iD e −1 (x, y) − i∆ ab
(91)
eab (x, y) we write D eab (x, y) as a Fourier integral to get the inverse of the matrix D eab (x, y) = D
where
Z
d4 q e iq(x−y) 4 Dab (q) e (2π)
→ → eab (q) = m2 + q 2 δab − 1 θ2 δab m2 + q 2 m2 + − D q 2 − iθεab m2 + − q 2 q0 4
(92)
(93)
Now using these relations and the fact that eq(91) can be rewritten as
one can deduce that
Z
e ab (x, y) D ebc (y, z) = δac δ 4 (x, z) d4 y ∆ e ab (x, y) = ∆
with e ab (q) = D e −1 (q) = ∆ ab
Z
d4 q e iq(x−y) 4 ∆ab (q) e (2π)
(94)
(95)
2 → → → 1 2 m2 + − q 2 2 (m2 + − q 2) 1 m2 + − q2 2 δ + θ δ +θ δ (q ) +iθε q0 ab ab ab 0 ab m2 + q 2 4 m2 + q 2 (m2 + q 2 )3 (m2 + q 2 )2 (96)
it is easy to show that the free noncommutative propagator has the following symmetry properties
e ba (−q) e ab (q) = D e −1 (q) = δab ∆ e (q) + εab ηe (q) = δab ∆ e (q) + θεab η (q) = ∆ ∆ ab → →2 2 − 2 1 1 2 m2 + − q2 2 2 (m + q ) e e (q) = ∆ + θ + θ 3 (q0 ) = ∆ (−q) 2 2 m2 + q 2 4 m2 + q 2 (m + q ) → m2 + − q2 ηe (q) = θη (q) = iθ q0 = − ηe (−q) (m2 + q 2 )2 4.2.
(97) (98) (99)
Noncommutative Exact Propagator
e ′ (x, y) is given by The exact noncommutative propagator ∆ ab RY b dϕa (x) [ϕa (x) ϕb (y)] eiS[ϕ] e ′ (x, y) = h0| T {Φa (x) Φb (y)} |0i = − i∆ ab 16
a,x
RY a,x
b dϕa (x) eiS[ϕ]
(100)
In order to evaluate the functional integral eq(100), we make a Taylor expansion in powers of g b iS[ϕ]
e
# 5 X b(0) = 1 + i Sb(k) [ϕ] eiS [ϕ] "
(101)
k=1
Using this Taylor expansion in eq(100) and expanding the denominator by the binomial theorem, we obtain the following expression for the exact noncommutative propagator
e′ −i∆ ab
i 1 i i (3) (1) (2) (4) (5) 2 e ab (x, y)− gI (x, y)+ θgI (x, y)+ gθ I (x, y) − I (x, y) − I (x, y) (x, y) = −i∆ ab ab ab 4 ab 2 4 2 ab (102)
with
(1) Iab (2) Iab (3) Iab
(x, y) =
(x, y) = (x, y) =
(4)
Iab (x, y) =
Z
Z
Z
Z
4
d τd
4
dz
Z
Y
(c) a,x
4
(1) zKcd
4 ′ 4
4
(τ, z)
d z d zd τ K
b(0) [ϕ]
dϕa (x) [ϕa (x) ϕb (y)] [ϕc (z) ϕc (z) ϕd (z) ϕd (z)] eiS Z
Y
(c) a,x
(2)
′
(z ; z, τ )
d4 z ′ d4 zd4 τ K(3) (z ′ ; z, τ )
b(0) [ϕ]
dϕa (x) [ϕa (x) ϕb (y)] [ϕc (τ ) ϕd (z) ϕm (τ ) ϕm (τ )] eiS
Z
Z
Y b(0) dϕa (x) [ϕa (x) ϕb (y)] [ϕc (z ′ ) ϕc (z) ϕd (τ ) ϕd (τ )] eiS [ϕ]
(c) a,x
Y b(0) dϕa (x) [ϕa (x) ϕb (y)] [ϕc (z) ϕc (τ ) ϕd (z ′ ) ϕd (z ′ )] eiS [ϕ]
(c) a,x
and (5) Iab
(x, y) = εmc εld
Z
d4 z ′ d4 zd4 τ K(3) (z ′ ; z, τ ) × Z Y b(0) dϕa (x) [ϕa (x) ϕb (y)] [ϕm (τ ) ϕc (z ′ ) ϕl (z) ϕd (z ′ )] eiS [ϕ] (c) a,x
where the subscript (c) being added to the functional integral us that
(k) Iab (x, y) ,
Y dϕa (x) ... to remind (c)
R
a,x
k = 1, 2, 3, 4, 5, are connected Green’s functions ( connected Green’s
functions are obtained by disregarding all terms which factorize into two or more functions with no overlapping arguments). (k)
The connected Green’s functions Iab (x, y) , k = 1, 2, 3, 4, 5, can now be calculated with the help of the identity eq(90), direct but lengthy calculations lead to the following expressions for the connected Green’s functions 17
(1) Iab
(2) Iab
(3) Iab
(x, y) = 16i
Z
d4 q e iq(x−y) e 4 ∆ac (q) ∆cb (q) e (2π)
Z
d4 p e ∆ (p) (2π)4
Z
Z i d4 q h e d4 p e iq(x−y) e (x, y) = −8 ∆ (p) ∆ (q) ε ∆ (q) q e ac cd db 0 (2π)4 (2π)4 Z Z i d4 p d4 q h e iq(x−y) e p0 ηe (p) ∆ (q) ∆ (q) e +8 ac cb (2π)4 (2π)4
Z i iq(x−y) d4 p e d4 q h e → − 2 2 e ∆ (p) ∆ (q) ∆ (q) m + q e (x, y) = 8i ac cb (2π)4 (2π)4 Z Z i d4 q h e d4 p →2 ∆ − iq(x−y) 2 e e (p) + 8i ∆ (q) ∆ (q) e ac cb 4 4 m + p (2π) (2π) Z
(4) Iab
(5) Iab
Z
Z i d4 q h e d4 p e 2 iq(x−y) e (x, y) = 4i ∆ (q) ∆ (q) (q ) e ∆ (p) ac cb 0 (2π)4 (2π)4 Z Z i d4 q h e d4 p 2 e iq(x−y) e + 4i ∆ (q) ∆ (q) e ac cb 4 4 (p0 ) ∆ (p) (2π) (2π) Z i d4 p e d4 q h e 2 iq(x−y) e ∆ (q) ∆ (q) (q ) e ∆ (p) (x, y) = 2i ac cb 0 (2π)4 (2π)4 Z Z i d4 q h e d4 p 2 e iq(x−y) e + 2i ∆ (q) ∆ (q) e ac cb 4 4 (p0 ) ∆ (p) (2π) (2π) Z
(103)
(104)
(105)
(106)
(107)
Now, from eqs(103) − (107) one can easily show that e ′ (x, y) = ∆ ab
with
where Π∗(Loop)ab
Z
d4 q e ′ iq(x−y) 4 ∆ab (q) e (2π)
h i i h e ′ (q) = −i∆ e ab (q) + −i∆ e ac (q) iΠ∗ e db (q) − i∆ (q) −i ∆ ab (Loop)cd
(108)
(109)
Z i d4 p h d4 p e e ∆ (p) + 4θg q0 εab ∆ (p) − δab p0 ηe (p) (110) (q) = 4igδab (2π)4 (2π)4 Z d4 p → →2 − 2 m2 + − − 2 2 e (p) p 2 + (q0 )2 + (p0 )2 ∆ + iθ gδab 4 −2 m + q (2π) Z
18
5.
RENORMALIZED NONCOMMUTATIVE EXACT PROPAGATOR 5.1.
Renormalized Noncommutative Action
In order to renormalize the noncommutative complex scalar field we proceed as in ϕ4 − theory of a real self-interacting scalar field, that is we split the noncommutative action Sb [ϕB ] , expressed in terms of bare couplings mB , gB and bare fields ϕB , in a part depending
on the renormalized parameters m, g and field ϕ, and in a counter term part [41]-[44] [36] [45] [39]
Sb [ϕB ] =
Z
b = Sb(0) [ϕB ] + Sb(1) [ϕB ] + Sb(2) [ϕB ] + Sb(3) [ϕB ] + Sb(4) [ϕB ] + Sb(5) [ϕB ] (111) d4 xL B B B B B B
The bare quantities, such as ϕB , mB , and gB , are just chosen so that they cancel the di-
vergences and leave us with the desired renormalized quantity. The physical or renormalized scalar field ϕ, mass m and coupling constant g are defined by
1
ϕ=Z − 2 ϕB
, m2 = m2B + δm2
−1
g = Zg 2 g B
, Zg =
(1 + B) Z2
(112) (113)
with Z and δm2 to be chosen so that the propagators of the renormalized fields have poles in the same position and with the same residues as the propagators of the free fields in the absence of interactions. In this paper we are interested only to the propagator and its renormalization up to first order in the parameter g, so we can assume that ϕ= ϕB
, g = gB
, Z = Zg = 1
m2 = m2B + δm2
(114) (115)
(0)
The noncommutative action SbB [ϕB ] may then be written in terms of renormalized quan-
tities, as
where
1 (0) SbB [ϕB ] = − 2
Z
e (B) (x, y) ϕb (y) d4 xd4 yϕa (x) D ab 19
(116)
e (B) (x, y) = D (B) (x, y) − 1 θ2 M(B) (x, y) − θN (B) (x, y) D ab ab ab ab 4 (B)
(B)
(117)
(B)
using the expressions of Dab (x, y) , Mab (x, y) and Nab (x, y)
(B)
(R)
Dab (x, y) = Dab (x, y) − δm2 δ 4 (x, y) δab (118) → − (B) (R) Mab (x, y) = Mab (x, y) − δm2 m2 − δ 4 (x, y) δab − δm2 m2 − ∇ 2 δ 4 (x, y) δab (119)
(B)
(R)
Nab (x, y) = Nab (x, y) − δm2 ∂t δ 4 (x, y) εab
(120)
e (B) (x, y) as one can write D ab e (B) (x, y) = D e (R) (x, y)−δm2 δ 4 (x, y) δab + 1 θ2 δm2 D ab ab 4
h
i → − m2 − δ 4 (x, y) + m2 − ∇ 2 δ 4 (x, y) δab + θδm2 ∂t δ 4 (x, y) εab
e (R) (x, y) is given by eq(78) where D ab
e (R) (x, y) = D (R) (x, y) − 1 θ2 M(R) (x, y) − θN (R) (x, y) (121) D ab ab ab ab 4 Z iq(x−y) 4 d4 q (R) 2 2 2 m + q e (122) Dab (x, y) = δab m − δ (x, y) = δab (2π)4 Z 2 − 2 − →2 4 d4 q (R) 2 2 2 Mab (x, y) = δab m − m − ∇ δ (x, y) = δab m +→ q 2 eiq(x−y) 4 m +q (2π) → − (R) Nab (x, y) = m2 − ∇ 2 εab ∂t δ 4 (x, y) = εab
Z
(123)
d4 q →2 [iq ] eiq(x−y) = N (y, x) − 2 0 ba 4 m + q (2π)
(124)
(0) hence SbB [ϕ] can be rewritten as (0) SbB
1 [ϕ] = − 2
Z
e (R) (x, y) ϕb (y) + ∆Sb(0) [ϕ] d4 xd4 yϕa (x) D ab
where the counter term part ∆Sb(0) [ϕ] is given by
20
(125)
Z 1 2 d xd yϕa (x) δ (x, y) δab ϕb (y)− θδm d4 xd4 yϕa (x) ∂t δ 4 (x, y) εab ϕb (y) 2 Z 1 − θ2 δm2 d4 xd4 yϕa (x) m2 − δ 4 (x, y) δab ϕb (y) 8 Z h i → − 1 2 2 d4 xd4 yϕa (x) m2 − ∇ 2 δ 4 (x, y) δab ϕb (y) − θ δm 8
1 ∆Sb(0) [ϕ] = δm2 2
Z
4
4
4
The interaction part of the noncommutative action Sb [ϕB ] may then be written in terms
of renormalized quantities as
(k) (k) SbB [ϕB ] = SbR [ϕ] + O g 2
(k) where SbR [ϕ] are given by eq(15) − (19)
k = 1, 2, 3, 4, 5
(126)
Now eq(125) and eq(126) lead to the following expression for the noncommutative action
Sb [ϕB ]
(0) (1) (2) (3) (4) (5) Sb [ϕB ] = SbR [ϕ] + ∆Sb(0) [ϕ] + SbR [ϕ] + SbR [ϕ] + SbR [ϕ] + SbR [ϕ] + SbR [ϕ]
(127)
e (R) (x, y) is given by The renormalized noncommutative exact propagator ∆ ab e (R) (x, y) = h0| T {Φa (x) Φb (y)} |0i = − i∆ ab
RY b dϕa (x) [ϕa (x) ϕb (y)] eiSB [ϕ] a,x
RY dϕa (x) eiSbB [ϕ]
(128)
a,x
making a Taylor expansion in powers of g, we get
(R)
e (x, y) = −i∆ ab
# " 5 X RY b(0) (k) dϕa (x) [ϕa (x) ϕb (y)] 1 + i∆Sb(0) [ϕ] + i SbR [ϕ] [ϕ] eiSR [ϕ] a,x
k=1
"
5 X
RY dϕa (x) 1 + i∆Sb(0) [ϕ] + i a,x
k=1
(k) SbR
#
(129)
(0) iSbR [ϕ]
[ϕ] [ϕ] e
The functional integrals in eq(129) can now be performed with the help of the identity e (R) (x, y) can be written as eq(28), the renormalized noncommutative exact propagator ∆ ab
21
Z (R) ′ 2 e ac (x, z) ∆ e cb (z ′ , y) e e dzdz ′ δ 4 (z, z ′ ) ∆ − i∆ab (x, y) = −i∆ab (x, y) − iδm Z 2 e ac (x, z) εcd∆ e db (z ′ , y) ∂t δ 4 (z, z ′ ) dzdz ′ ∆ + iθδm Z i 2 2 e ac (x, z) ∆ e cb (z ′ , y) m2 − δ 4 (z, z ′ ) dzdz ′ ∆ + θ δm 4 Z i 2 2 →2 e e cb (z ′ , y) m2 − − + θ δm dzdz ′ ∆ ∇ ∆ac (x, z) 4 e ′ (x, y) is given by eqs(108) − (110) . where −i∆ ab e (R) (x, y) as a Fourier integral If we write ∆ ab
e (R) ∆ ab
then
where
d4 q e (R) iq(x−y) 4 ∆ab (q) e (2π)
(130)
e (R) (q) = ∆ e ′ (q) + δm2 ∆ e ac (q) ∆ e cb (q) − iθδm2 q0 ∆ e ac (q) εcd ∆ e db (q) ∆ (131) ab ab 1 → e ac (q) ∆ e cb (q) − 1 θ2 δm2 m2 + − e ac (q) ∆ e cb (q) − θ2 δm2 m2 + q 2 ∆ q2 ∆ 4 4
e ′ (q) = ∆ e ab (q) + ∆ e ac (q) Π∗ e ∆ ab (Loop)cd (q) ∆db (q) =
and Π∗(Loop)ab
(x, y) =
Z
(q) = 4igδab
Z
+ iθ2 gδab
Z
(132)
Z Z d4 p e d4 p d4 p e 2 ∆ (p) + 4θgεab ∆ (p) q0 − 4θ gδab p0 η (p) (2π)4 (2π)4 (2π)4 (133)
d4 p →2 − 2 m2 + − − → 2 e (p) p 2 + (q0 )2 + (p0 )2 ∆ 4 −2 m + q (2π)
from eq(131) and eq(132) we get
e (R) (q) = ∆ e ab (q) + ∆ e ac (q) Π∗ (q) ∆ e db (q) = ∆ cd ab
"
1 e −1 (q) − Π∗ (q 2 ) − iǫ ∆
where Π∗ab (q) is the self-energy part (of the propagator) 22
#
ab
+ O g2
(134)
1 1 → q 2 δab + Π∗(Loop)ab (q) Π∗ab (q) = δm2 δab −iθδm2 q0 εab − θ2 δm2 m2 + q 2 δab − θ2 δm2 m2 + − 4 4 (135) Using the expression eq(133) of Π∗(Loop)ab (q), the self-energy part Π∗ab (q) can be rewritten as
Π∗ab
h i 1 2 2 (2) (1) 2 (q) = 1 − θ m δm δab + πab − iθδm2 εab − πab q0 2 1 2 2 1 2 2 (3) − (4) → 2 − θ δm δab − πab q + θ δm δab − πab (q0 )2 2 4
(136)
where (1) πab
(2) πab
= 4gi
= 4gθ
Z
Z
Z 1 2 1 2 d4 p d4 p 2 e → − 2 2 2 p0 η (p) δab 1 − θ 2m + p + θ (p ) ∆ (p) δ − 4gθ 0 ab 2 4 (2π)4 (2π)4
d4 p e (3) 2 4 ∆ (p) εab , πab = −2giθ (2π)
Z
d4 p e ∆ (p) (2π)4
,
(4) πab
= giθ
2
Z
(137)
d4 p e ∆ (p) (2π)4 (138)
the last three terms of eq(136) can be written as Z i h d4 p e (2) 2 2 ∆ (p) εab q0 iθδm εab − πab q0 = iθ δm + 4gi (2π)4 Z 1 2 2 d4 p e 1 2 (3) − →2 − → 2 2 θ δm δab − πab q = θ δm + 4gi 4 ∆ (p) δab q 2 2 (2π) Z 1 2 2 1 2 d4 p e (4) 2 2 2 θ δm δab − πab (q0 ) = θ δm + 4gi 4 ∆ (p) (q0 ) 4 4 (2π)
(139) (140) (141)
while the first term of eq(136) reads
Z d4 p e 1 2 2 1 2 2 (1) 2 2 1 − θ m δm δab + πab = δm + 4gi ∆ (p) δab − θ m 2 2 (2π)4 Z Z 4 dp 1 d4 p 2 e → − 2 2 2 + 2giθ p0 η (p) δab −m − p + (p0 ) ∆ (p) + 2gi 2 (2π)4 (2π)4
Putting δm2 = δm20 + 21 θ2 δµ20 in eq(136) , we get 23
Π∗ab
Z 2 (q) = δm0 + 4gi
where
1 2 2 d4 p ∗ ∗ δµ0 δab + ̟ab 4 ∆ (p) πab (q) + θ 2 (2π)
∆ (p) =
m2
1 + p2
(142)
(143)
∗ ∗ is the free-field propagator, while πab (q) and ̟ab are given by
∗ πab
1 1 → (q) = δab − iθεab q0 − θ2 δab m2 + − q 2 + θ2 δab (q0 )2 2 4 ∗ ̟ab
5.2.
= 8giδab
Z
→ m2 + − p2 d4 p 1 4 (p0 ) − + 4 (m2 + p2 )3 (2π)4
(144)
(145)
Dimensional Regularization
To deal with the divergent integrals in eq(142) and eq(145), one can use some sort of regularization technique that makes these integrals finite. Dimensional regularization is the most powerful and popular method of regulating divergent integrals in perturbation theory. the idea is to treat the loop integrals as integrals over D-dimensional momenta, then we could analytically continue the integrals back to D = 4. The ultraviolet divergences will then appear as singularities (poles) in the deviation, ǫ = 4−D, from four dimensions [41]-[44] [36] [40] [45]. Let us consider the divergent integral I1 = 4gi
Z
d4 p ∆ (p) (2π)4
(146)
The first step in doing this divergent integral is to analytically continue the Feynman integrals to a continuous space-time dimension in the neighborhood of the physical dimension D=4 I1 = 4giµ
4−D
Z
dD p (2π)
D
m2
1 + p2
(147)
where µ is an arbitrary mass parameter introduced to keep the bare coupling constant g dimensionless.
24
The next step is called a Wick rotation [36] [42] [44], instead of integrating p0 on the real axis from −∞ to +∞, we integrate it on the imaginary axis from −i∞ to +i∞. That is, → → we can write p0 = iq4 , p2 = − p 2 − (p0 )2 = − q 2 + (q4 )2 = q 2 , with q4 integrated over real values from −∞ to +∞
I1 = 4giµ
4−D
Z
dD p
1 = −4gµ4−D D 2 2 m + p (2π)
Z
dD q (2π)
D
m2
1 + q2
(148)
Using the identity [46] [38] [42] [44] −z
a
1 = Γ (z)
Z
∞
dttz−1 e−at
(149)
0
which is valid for Re z > 0 and Re a > 0, where Γ (z) is the Gamma function Z ∞ Γ (z) = dttz−1 e−t
(150)
0
defined for Re(z) > 0, we obtain the following expression I1 = −4gµ
4−D
Z
µ4−D 1 = −4g (2π)D m2 + q 2 (2π)D dD q
Z
∞
dt
0
Z
dD qe−(m
2 +q 2
)t
(151)
integration over the D-dimensional momentum integral can be performed by means of the relation
Z
dD qe−(m
2 +q 2
D )t = π 2 e−m2 t t
(152)
Then we can perform the integral over t using the formula Z +∞ α−s dtt2s−1 e−αt = Γ (s) 2 0
(153)
yielding to
D
I1 = −4g
µ4−D π 2 (2π)D
2
m
D2 −1
Z
0
∞
−D −t 2
dtt
e
m2 = −4g 16π 2
m2 4πµ2
D2 −2 D Γ 1− 2
(154)
The divergence of this integral manifests itself in the pole of the Gamma function at the physical dimension D = 4. In the neighborhood of this dimension we set D = 4 − 2s then 25
(155)
gm2 I1 = − 2 4π
−s
m2 4πµ2
Γ (s − 1)
(156)
The singularity at small s → 0 may be isolated by using the expansions Γ (s − 1) = − and
we get then I1 = 4giµ
4−D
Z
m2 4πµ2
−s
1 − γ + 1 + O (s) s
m2 = 1 − s ln 4πµ2
+ O (s)
2 1 gm2 1 gm2 m = + 1 − γ − ln D 2 2 2 2 4π s 4π 4πµ2 (2π) m + p dD p
(157)
(158)
(159)
Let us now evaluate the divergent integral eq(145)
I2 = 8gi
Z
→ m2 + − p2 1 d4 p 4 − + (p0 ) 4 (m2 + p2 )3 (2π)4
(160)
dimensional regularization gives Z Z → → dD q 1 1 m2 + − p2 m2 + − q2 dD p 4 4 4−D 4−D (p0 ) = −8gµ (q4 ) − + − + I2 = 8giµ 4 (m2 + p2 )3 4 (m2 + q 2 )3 (2π)D (2π)D (161) where we have performed the usual Wick rotation. Next we use the relation eq(149) 1 1 3 = 2 2 Γ (3) (m + q )
Z
∞
dtt2 e−(m
2 +q 2
)t
(162)
0
and the fact that [44] Z
to get
I2 = −8gµ
4−D
Z
dD q (2π)D
=0
(163)
Z ∞ Z → 2 2 dD q dD q m2 + − q2 µ4−D 4 4 → − 2 2 dtt2 e−(m +q )t m + q (q4 ) (q4 ) = −8g 3 D D Γ (3) (2π) (m2 + q 2 ) (2π) 0 (164) 26
I2 can be rewritten as
I2 = −4g
µ4−D (2π)
D
Z
∞
2 −m2 t
dtt e
0
Z
d
− →2 − q m +→ q 2 e− q t
→ D−1 −
2
Z
+∞
2
dq4 (q4 )4 e−q4 t
(165)
−∞
the integration over the D-dimensional momentum integral on the right-hand side can be performed with the help of the relations [44] [46] [38] [42] Z n Z +∞ 2π 2 n 2 dkk n−1 f q 2 d qf q = n Γ 2 0
(166)
Z
α−s Γ (s) 2
(167)
Z ∞ Z +∞ 5 2 − 12 −m2 t dtt e dkk D−2 m2 + k 2 e−k t 2 0 0
(168)
and
hence
D−1
µ4−D 2π 2 Γ I2 = −4g (2π)D Γ D−1 2
using eq(167), we get
I2 = −g
3µ4−D (2π)D
+∞
dtt2s−1 e−αt = 0
π
D 2
2
m
D2
Z
∞
0
D − 1 − D+2 −t −D t 2 e dt t 2 + 2
(169)
using again eq(167) to perform the integration over t, leads to 3m4 I2 = −g 16π 2
m2 4πµ2
D2 D−1 D D Γ − Γ − +1 + 2 2 2
(170)
The divergence of this integral manifests itself in the pole of the Gamma function at the physical dimension D = 4. In the neighborhood of this dimension we set D = 4 − 2s then 3m4 I2 = −g 16π 2 using the expansions
m2 4πµ2
−s 3 Γ (s − 1) + − s Γ (s − 2) 2 27
(171)
1 Γ (s − 1) = − − γ + 1 + O (s) s 3 1 1 + O (s) −γ+ Γ (s − 2) = 2 s 2
and
(172) (173)
−s 2 m2 m + O (s) = 1 − s ln 2 4πµ 4πµ2 we get the following expression
(174)
2 3 m 3gm4 1 − γ + − ln I2 = 2 64π s 2 4πµ2
(175)
Hence
∗ ̟ab
= 8giµ
4−D
δab
Z
dD p (2π)D
2 → 1 m2 + − p2 3 1 m 3gm4 4 − + δab − γ + − ln (p0 ) = 3 2 4 (m2 + p2 ) 64π s 2 4πµ2 (176)
and
Π∗ab
where
2 gm2 1 gm2 m 2 ∗ (q) = δm0 + + 1 − γ − ln πab (q) 2 2 4π s 4π 4πµ2 2 3gm4 1 3gm4 3 m 1 2 2 δab + − γ − ln + θ δµ0 + 2 64π 2 s 64π 2 2 4πµ2
(177)
1 2 1 2 2 → − 2 2 (q) = δab − iθεab q0 − θ δab m + q + θ δab (q0 ) (178) 2 4 The condition that m2 is the true mass of the particle is that the pole of the propagator ∗ πab
should (like the uncorrected propagator) be at q 2 = −m2 , so that − → [Π∗ab (q)]q2 =−m2 = Π∗ab 0 , m = 0
This condition allows us to evaluate δm20 and δµ20 2 gm2 1 gm2 m 2 δm0 = − 2 − 1 − γ − ln 2 4π s 4π 4πµ2 2 m 3gm4 1 3gm4 3 2 − − γ − ln δµ0 = − 2 2 64π s 64π 2 4πµ2
(179)
(180) (181)
to this order Π∗ (q) is equal to zero
Π∗ (q) = 0 + O g 2 28
(182)
6.
CONCLUSION
Thought this work we have considered a noncommutative complex scalar field theory with self interaction, by imposing non commutativity to the canonical commutation relations. The action and all relevant quantities are expanded up to second order in the parameter of noncommutativity θ. Using the path integral formalism, the noncommutative free and exact propagators are calculated to one-loop order and to the second order in the parameter of noncommutativity θ. Dimensional regularization was used to remove ultraviolet divergences that arise from loop graphs. It has been shown that these divergences may also be absorbed into a redefinition of the parameters of the theory.
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