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IL NUOVO CIMENTO VOL. 110 A, N. 11 Novembre 1997

A new regularization method in 3-dimensional momentum space (*)

LIANG-GANGLIU(1)(2), XIANG-QIANLUO(1)(2) and WEICHEN(2) (1_{) CCAST (World Laboratory) - P.O. Box 8730, Beijing 100080, PRC} (2_{) Department of Physics, Zhongshan University - Guangzhou 510275, PRC} (ricevuto l’11 Aprile 1997; approvato il 27 Novembre 1997)

Summary. — We propose a new method to calculate the 4-dimensional divergent

integrals. By calculating the one-loop integral as an example, the regularization of the integrals in 3-dimension momentum space is given in detail. We find that the new method gives the same results as the traditional dimensional regularization method, but it has the advantage of giving the real and the imaginary part separately. PACS 11.10.Gh – Renormalization.

1. – Introduction

It is well known that the dimensional regularization method is a powerful and elegant tool to calculate and regularize the divergent integrals in 4-dimensional energy-momentum space [1, 2]. This method is widely used both in particle physics and in nuclear physics [3, 4] to calculate the loops integration and renormalization. But in some cases, such as of the zero-point fluctuation energy

s

Qdp

k

p

2

1 m2, where m is the mass of nucleon or mesons, cannot be calculated by using the dimensional regular-ization formulae directly. In this case, a cut-off momentum is introduced to truncate the upper limit of the integration to make the divergence controllable [5]. In our studies, we found that this kind of 3-dimensional divergent integrals can be regularized by a new method [6]. In this paper, we will demonstrate that this method is also applicable to calculate other divergent integrals in 4-dimensional energy-momentum space.

In the next section, we will show briefly the traditional dimensional regularization of the integrals F1and F2. In sect. 3, we give the details of the calculation of F1, F2by

our new method. A summary is given in the last section.

(*) The authors of this paper have agreed to not receive the proofs for correction.

LIANG-GANG LIU,XIANG-QIAN LUO,WEI CHEN

1278

2. – 4-dimensional regularization of F1and F2

In the one-loop level of vacuum fluctuation, the polarization insertion to the vertex function or propagators can be expressed in term of two functions [7]:

F14 i ( 2 p)4



Q d4_q 1 q2 2 m21 ie , (1) F2(k2, m12, m22) 4 i ( 2 p)4



Q d4_q 1 (q2_{2 m}121 ie)[ (q 2 k)22 m221 ie] , (2)

here k , q are four energy-momentum (1_{). The subscript Q indicates that the} integra-tion is in the whole 4-dimensional space. By use of the dimensional regularizaintegra-tion technique [1, 2, 3], F1and F2are readily calculated, the results are follows:

F14 2 m2 16 p2

(

G(e) 2ln m 2 1 1 1 O(e)

)

, (3) F2(k2, m12, m22) 42 1 16 p2

u

G(e) 2



0 1 dx ln Mk2(x) 1O(e)

v

, (4) Mk2(x) 4k2x22 (k21 m122 m22) x 1m12, (5)

where e K01_{, G(e) is gamma function, G(e) A (1Oe) so F}

1, F2 are divergent. Notice

that M2

k(x) is not positive-definite, provided that k2D (m11 m2)2, which is the

threshold at which the incoming particle can decay into particles m1, m2. Thus a

imaginary part of F2appears,

Im F2(k2, m12, m22) 42 1 16 p kD k2 u(k 2 D

(

m11 m2)2

)

, (6) D 4 [k22 (m11 m2)2] [k22 (m12 m2)2] . (7)

3. – Regularization in 3-dimensional space

We propose in the following a new regularization method in 3-dimensional space. Taking F1 and F2 as examples to demonstrate that the results are the same as

above.

3.1. Regularization of F1. – The function F1in eq. (1) can be expressed as (q02 Eq1

ie)(q01 Eq2 ie), where Eq4

k

q21 m2. By use of Cauchy theorem, the integration of

q0in the complex q0plane gives

F14 1 2( 2 p)3



Q dq Eq . (8)

(1_{) We follow the convention of B}

JORKEN J. D. and DRELL S. D., Relativistic Quantum Fields (McGraw-Hill, New York) 1965.

A NEW REGULARIZATION METHOD IN3-DIMENSIONAL MOMENTUM SPACE 1279

Now we assume that the 3-dimensional integration is performed in n dimensions with the limit n K3, so dqKdn

q 4qn 21_{dq dV}

n, with

s

QdVn4 4 p in the limit n K 3,

here q f NqN. Then eq. (8) can be reduced as follows:

F14 1 8 p2(m 2₎(n23)O2



0 Q dt tnO221_{( 11t)}21O2 4 1 8 p2(m

2₎(n23)O2 G(nO2) G(1O22nO2)

G_{( 1 O2)} . (9)

Expanding it in a Laurent series about e 4 (32n)O2 gives the same result as in eq. (3).

3.2. Regularization of F2(k2, m12, m22). – F2(k2, m12, m22) can be rewritten as

follows: F2(k2, m12, m22) 4 i ( 2 p)4



0 1 dx



Q dn_{q dq} 0 1 [q2 02 q22 Mk2(x) ]2 . (10)

Since Mk2(x) can be both positive and negative, we should locate the poles in the

denominator of eq. (11). The analysis shows that

.

`

/

`

´

M2 k(x) D0 M2 k(x) E0 for or or for k2_{E (m}11 m2)2, k2 D (m11 m2)2 and 0 ExEx1, k2 D (m11 m2)2 and x2E x E 1 , k2 D (m11 m2)2 and x1E x E x2, (11) where x1 , 24 1 2 k2(k 2 1 m122 m22ZkD) . (12)

Then eq. (11) can be written as follows:

F2(k2, m12, m22) 4 i ( 2 p)4

{

u[k 2 E (m11 m2)2]



0 1 dx



Q dn_{q dq} 0 1

(

q2 02 Eq(x)21 ie

)

2 1 (13) 1 u[k2D (m11 m2)2]

u



0 x1 dx 1



x2 1 dx

v



Q dn_{q dq} 0 1

(

q022 Eq(x)21 ie

)

2 1 1 u[k2D (m11 m2)2]



x1 x2 dx



Q dn_{q dq} 0 1

(

q2 02 q2(x) 1NMk2(x) N

)

2

}

, where Eq(x)24 q21 Mk2(x). It can be verified that



Q dn_{q dq} 0 1

(

q2 02 Eq(x)21 ie

)

2 4



Q dn_{q dq} 0 1

(

q2 02 q2(x) 1NMk2(x) N

)

2 4 (14) 4 ip2

(

G(e) 2ln Mk2(x) 1O(e)

)

.

LIANG-GANG LIU,XIANG-QIAN LUO,WEI CHEN

1280

Substituting eq. (14) into eq. (13) we obtain

F2(k2, m12, m22) 42 1 16 p2

u

G(e) 2



0 1 dx ln NM2 k(x) N1O(e)

v

1 (15) 1 i ( 2 p)4 Q (2ip 2_{) Q (x} 22 x1) Q ipu[k2D (m11 m2)2] 4 4 2 1 16 p2

u

G(e) 2



0 1 dx ln NMk2(x) N

v

2 i 16 p kD k2 u[k 2 D (m11 m2)2] 1O(e) .

We can see that it is the same as eq. (4) and eq. (6), therefore, we demonstrate that 3-dimensional regularization gives the same results as the traditional 4-dimensional regularization method. This method can be extended to calculate other divergent integrals, such as zero-point fluctuation energy mentioned in the introduction. The calculation gives 2 1 ( 2 p)3



Q dq

k

q2 1 m2K 2 1 ( 2 p)3



Q dn_q

_k

_q2 1 m24 (16) 4 2 1 32 p2m 4

g

_G (e) 2ln m2 1 3 2

h

1 O(e) , which is again the same as that given in ref. [8].

4. – Conclusions

We have demonstrated by calculating integrals F1 and F2 that the new

3-dimensional regularization method gives the same results as those of the traditional 4-dimensional regularization method. We also show that the new method can be used to calculate the zero-point fluctuation energy of relativistic particles. In fact, this method can be generalized further to evaluate the quantum effect in two- or one-dimensional systems and has a wide application perspective in physics.

R E F E R E N C E S

[1] ’THOOFTG. and VELTMANM., Nucl. Phys. B, 44 (1972) 189.

[2] RAMONDP., Field Theory: A Modern Primer (Addison-Wesley, Redwood City) 1981. [3] SEROT B. D. and WALECKA J. D., Recent progress in quantum hadrodynamics, to be

published in the Int. J. Mod. Phys. E.

[4] LIUL. G., BENTZW. and ARIMAA., Ann. Phys., 194 (1989) 387. [5] NAKANOM. et al., Phys. Rev. C, 55 (1997) 1.

[6] LIUL. G., Quantum Effects in Nuclear Matter, thesis, University of Tokyo PhD (1989). [7] MIGNACOJ. A. and REMIDDIE., Nuovo Cimento A, 1 (1971) 376.