UV IR phenomenon of noncommutative quantum fields in example
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VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
UV/IR phenomenon of Noncommutative Quantum
Fields in Example
Nguyen Quang Hung*, Bui Quang Tu
Faculty of Physics, VNU University of Science, 334 Nguyễn Trãi, Hanoi, Vietnam
Received 05 December 2014
Revised 18 February 2015; Accepted 20 March 2015
Abstract: Noncommutative Quantum Field (NCQF) is a field defined over a space endowed with
a noncommutative structure. In the last decade, the theory of NCQF has been studied intensively,
and many qualitatively new phenomena have been discovered. In this article we study one of these
phenomena known as UV/IR mixing.
Keywords: Noncommutative quantum field theory.
1. Introduction∗
Noncommutative quantum field theory (NC QFT) is the natural generalization of standard
quantum field theory (QFT). It has been intensively developed during the past years, for reviews, see
[1,2]. The idea of NC QFT was firstly suggested by Heisenberg and the first model of NC QFT was
developed in Snyder’s work [3]. The present development in NC QFT is very strongly connected with
the development of noncommutative geometry in mathematics [4], string theory [5] and physical
arguments of noncommutative space-time [6].
The simplest version of NC field theory is based on the following commutation relations between
coordinates [7]:
[ xˆ µ , xˆν ] = i θ µν ,
where θ
µν
(1)
is a constant antisymmetric matrix.
Since the construction of NC QFT in a general case ( θ 0i ≠ 0 ) has serious difficulties with unitarity
and causality [8-10], we consider a simpler version with θ 0i = 0 (thus space-space noncommutativity
only), in which there do not appear such difficulties. This case is also a low-energy limit of the string
theory [1, 2].
_______
∗
Corresponding author. Tel.: 84- 904886699
Email:
47
N.Q. Hung, B.Q. Tu / VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
48
2. Moyal Product
We introduce d -dimensional noncommutative space-time by assuming that time and position are
not c -numbers but self-adjoint operators defined in a Hilbert space and obeying the commutation
algebra
[ xˆ µ , xˆν ] = i θ µν ,
(2)
µν
where the θ are the elements of a real constant d × d antisymmetric matrix θ . Then we define
the Moyal star product
n
∞
i 1
f ( x) g ( x) = f ( x) g ( x) + ∑ θ µ1ν1 …θ µnν n [∂ µ1 …∂ µn f ( x)] [∂ν1 …∂ν n g ( x)]
n =1 2 n!
i ∂ µν ∂
= f ( x) exp
θ
g ( x).
µ
∂xν
2 ∂x
(3)
In particular we have:
e
ipµ x µ
ν
i
i ( p + q )µ x µ
eiqν x = exp − p ∧ q e
,
2
(4)
where we have defined the wedge product
p ∧ q = ∑ pµθ µν qν .
(5)
µ ,ν
The natural generalization of the star product (3) follows:
f1 ( x1 )
i
∂ ∂
f n ( xn ) = ∏ exp θ µν µ ν f1 ( x1 )
∂xa ∂xb
a
f 2 ( x2 )
f n ( xn ), for a, b = 1,…, n. (6)
A simple prescription to construct NC FT is to replace ordinary products by (Moyal) star products
all over the place. For example, the action for a noncommutative Φ 4 real-valued scalar field
1
m2
λ
S [Φ ] = ∫ d d x ∂ µ Φ ∂ µ Φ −
Φ Φ − Φ Φ Φ Φ.
2
4!
2
(7)
For θ 0i = 0 we can construct NC quantum fields by canonically quantizing NC classical fields.
This can be done by applying formal canonical quantization method. Alternatively, we can quantize
NC classical fields by path intergral method. Thus
Z [ J ] = ∫ D µ [Φ ] e i S [ Φ ] e i ∫ d
d
x ( J Φ)
,
(8)
with some specification of the integral measure.
3. Noncommutative Perturbative Quantization
Now we will restrict ourselves to the pertubative evaluation of Z [ J ] . The first important
observation is that the free approximation is locally θ -independent
N.Q. Hung, B.Q. Tu / VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
S[Φ ]free =
1 d
∫ d x ∂ µ Φ ∂ µ Φ − m2Φ Φ = 1 ∫ d d x ∂ µ Φ∂ µ Φ − m2 Φ 2 .
2
2
49
(9)
The Fourier transform of the Feynman propagator is the same as for commutative scalar field
G( p) =
i
.
p − m2 + i 0
(10)
2
Upon Fourier transformation
∫d
d
x Φ ( x)
n
k =1
Φ ( x) = ∫ d d p (2π ) d δ ∑ pk Φ ( p1 )
where W ( p1 ,…, pn ) = exp −
Φ ( pn ) W ( p1 ,…, pn ),
i
,
p
∧
p
∑ i j
2 i< j
(11)
(12)
is the Moyal phase. Thus we get a simple Feynman rule for the interactions:
−iλn → −i λn W ( p1 ,…, pn ),
(13)
i.e. the standard Feynman vertex is mapped into itself times the Moyal phase.
Hence, the Feynman rules in momentum space of noncommutative field theory are similar to those
of commutative ones except that the vertices of the NC theory are modified by the Moyal phase factor.
4. The UV/IR mixing of NC QFT
The phenomenon of UV/IR mixing is the most radical feature of NC QFT that significantly differs
from those of ordinary QFT. It occurs in perturbation theory, so we can study this phenomenon in
details. We analyze the UV/IR mixing in the case of real-valued Φ 4 scalar field.
The NC real-valued Φ 4 theory in the four-dimensional space-time, is described by
1
m2
λ
L = ∂µΦ ∂µ Φ −
Φ Φ − Φ Φ Φ Φ.
2
2
4!
(14)
As we have seen in Eqs (9), (13), under the integration the star product of the fields does not affect
the quadratic parts of the Lagrangians, whereas it makes the interaction parts become nonlocal by the
Moyal phase (12).
For the Lagrangian (14), the Feynman rule for the noncommuative vertex is
−iλ 1
1
cos ( p1 ∧ p2 + p1 ∧ p3 + p2 ∧ p3 ) + cos ( p1 ∧ p2 + p1 ∧ p3 − p2 ∧ p3 ) +
3 2
2
1
cos ( p1 ∧ p2 − p1 ∧ p3 − p2 ∧ p3 ) ,
2
(15)
where pi , i = 1,…, 4 , are momenta coming out of the vertex and pi ∧ p j = piµθ µ ,ν p jν .
In the commutative Φ 4 model the leading mass renormalization comes from the normal-ordering
diagram contribution to the self energy [11]:
N.Q. Hung, B.Q. Tu / VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
50
Σ commutative = −
Λ2
m4
d 4k
1
−λ 2
2
=
Λ
−
m
ln
+
O
2
4
2
2
2
2
2 i (2π ) k + m
32π
m
Λ
λ
∫
,
(16)
where Λ is the ultraviolet cutoff.
In the noncommutative Φ 4 model, we have two contributions, planar and nonplanar Feynman
diagrams. The planar diagram gives almost the same contribution (16), except the factor 1 / 3 instead
of 1 / 2 , which is responsible for different symmetry of the diagram. Thus
Σ NCP = Σ nc planar = −
Λ2
d 4k
1
−λ 2
2
=
Λ
−
m
ln
2 +
3 i (2π ) 4 k 2 + m 2 48π 2
m
λ
∫
.
(17)
and the nonplanar diagram gives
Σ NCNP = Σ nc nonplanar = −
2
Λ eff
d 4 k cos( p ∧ k )
λ 2
2
=
−
Λ
−
m
ln
eff
m2
6 i (2π ) 4 k 2 + m 2
96π 2
λ
∫
+
(18)
where
p = pµθ µν and Λ 2eff =
1
p + 1 / Λ2
(19)
2
is the effective cutoff, which shows the mixing of UV divergence and IR singularity.
Note that the nonplanar contribution is one half of the planar one. We computed all above
integrations by using dimensional regularization method [11]. So we can normalize the theory at fixed
p and fixed θ by subtracting the planar divergence in the limit when the cutoff Λ tends to infinity
m 2 → M 2 = m2 −
Λ2
λ 2
2
Λ
−
m
ln
2 .
48π 2
m
(20)
Finally, we obtain one particle irreducible (or 1PI) effective action
Γ1PI = ∫ d 4 pΦ (− p )Γ (2) ( p )Φ ( p ) +
Γ (2) ( p ) = p 2 + M 2 −
(21)
λ
λM 2 1
ln
+
+
2
2
96π 2 p 48π 2 M 2 p
(22)
Thus the effective action has a singularity at p = 0 that can be interpreted either as a non-analytic
function of θ at fixed p , or an IR singularity at fixed θ .
In the case that Φ is a complex scalar field, there are two ways of ordering the fields Φ and Φ ∗
in the quartic interaction (Φ ∗Φ ) 2 . So, the most general potential of the NC complex scalar field action
is
V (Φ ) = AΦ ∗ Φ Φ ∗ Φ + BΦ∗
Φ ∗ Φ Φ.
(23)
It was shown in [12] that the theory is not generally renormalizable for arbitrary values of A and
B and is renormalizable at one-loop level only when B = 0 or A = B .
N.Q. Hung, B.Q. Tu / VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
51
5. Conclusion
Our main focus in this article is to point out several important aspects of NC field theories,
especially noncommutative perturbative path-integral quantization and the renormalization problem of
NC QFT. We have figured out significant analogies and radical differences between the perturbative
description of NC QFT and that of the ordinary QFT. We successfully calculated noncommutative
vertex, one-loop renormalized mass and 1PI effective action for noncommutative real-valued scalar
field. We found that UV/IR mixing terms, as a direct consequence of phase factors induced in the
vertex, generally appear in all perturbative quantum calculations. The analysis and computing
techniques used here are very useful and applicable for other models of NC QFT.
Acknowledgments
This work was partially supported by the Hanoi University of Science Grant No. TN-14-08.
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