Asymptotic freedom of non-Abelian tensor gauge fields

Physics Letters B 732 (2014) 150–155

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Physics Letters B www.elsevier.com/locate/physletb

Asymptotic freedom of non-Abelian tensor gauge fields George Savvidy Institute of Nuclear and Particle Physics, Demokritos National Research Center, Ag. Paraskevi, Athens, Greece

a r t i c l e

i n f o

Article history: Received 30 January 2014 Accepted 12 March 2014 Available online 19 March 2014 Editor: L. Alvarez-Gaumé

a b s t r a c t We calculated the one-loop contribution to the Callan–Symanzik beta function which is induced by the non-Abelian tensor gauge fields. The contribution is negative and corresponds to the asymptotically free theory. Two methods have been used to calculate the beta function, the first one based on generalization of the Altarelli–Parisi momentum sum rule, which implicitly comprises unitarity, and the second one – on the effective Lagrangian approach. Possible consequences are discussed. © 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .

1. Introduction In the 70ies it was discovered that the non-Abelian Yang–Mills fields are asymptotically free and that the Bjorken scaling should be broken by logarithms of a transverse momentum [1–8]. The exceptional physical properties of the non-Abelian gauge fields were rooted in the negativity of the corresponding Callan–Symanzik beta function. In this article we are interested to calculate the Callan– Symanzik beta function in the generalized YM theory [9–12]. The proposed generalization of the YM theory describes the interaction of the spin 1 – “gluons” with their massless partners of higher spin – “tensor-gluons”.1 The characteristic property of the model is that all interaction vertices between gluons and tensor-gluons have dimensionless coupling constants in four-dimensional space–time. That is, the cubic interaction vertices have only first order derivatives and the quartic vertices have no derivatives at all. These are familiar properties of the standard Yang–Mills field theory. In order to understand the physical properties of the model it was important to study the tree level scattering amplitudes. A spinor representation of the amplitudes was used to calculate high-order tree level diagrams with the participation of tensorgluons in [13]. They are the generalizations of the Parke–Taylor scattering amplitude [14] in the case of two tensor-gluons of spin s and (n − 2) gluons. These scattering amplitudes allow to derive the splitting functions of the gluons into tensor-gluons [15]. It is important to understand the behavior of the quantum corrections as well. The direct calculation of the Callan–Symanzik beta function encounters enormous complications because of the high-rank tensor structure of the vertices and propagators. But the

1

In the rest of the article we shall use the QCD terminology for certainty.

availability of the tree level splitting functions written in terms of the physical helicities of the interacting particles together with the generalization of the Altarelli–Parisi momentum sum rule [4], which implicitly comprises unitarity, lead to the desired result. A nontrivial regularization of infrared singularities is required to achieve the final result. We also use the effective Lagrangian approach to calculate the radiative corrections [29–31] and to get convinced that both methods give the identical results. The present paper is organized as follows. In the next section we shall derive the tensor splitting functions and in the third section the generalized evolution equations, which take into account the processes of emission of tensor-gluons by gluons. The generalized Altarelli–Parisi momentum sum rule allows to find the tensor-gluons contribution to the one-loop Callan–Symanzik beta function, which takes the following form:

α (t ) = b=

α 1 + bαt

,

(12s2 − 1)C 2 (G ) − 4n f T ( R ) 12π

,

s = 1, 2, . . . ,

(1.1)

where s is the spin of the gauge bosons. The matrix of anomalous dimensions for the twist-two operators γn can also be computed. Appendix A contains the details of the regularization scheme. 2. Splitting functions It is convenient to represent a scattering amplitude of the ˜ i and correspondmassless particles in terms of spinors λi , λ ing helicities h i [14,16–27]. In spinor representation one can derive a general form for the three-particle interaction vertices M 3 (1h1 , 2h2 , 3h3 ) [13,25] which has the dimensionality [mass] D =±(h1 +h2 +h3 ) .

http://dx.doi.org/10.1016/j.physletb.2014.03.022 0370-2693/© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .

G. Savvidy / Physics Letters B 732 (2014) 150–155

In the generalized Yang–Mills theory [9–12] all interaction vertices between high-spin particles have dimensionless coupling constants, which means that the helicities of the interacting particles in the vertex are constrained by the relation D = ±(h1 + h2 + h3 ) = 1. Using these vertices one can compute the scattering amplitudes of gluons and tensor-gluons. The color-ordered scattering amplitudes involving two tensor-gluons of helicities h = ±s, one negative helicity gluon and (n − 3) gluons of positive helicity were found in [13]:

  ˆ n 1+ , . . . , i − , . . . , k + s , . . . , j − s , . . . , n + M 2s−2   ˙ i j 4 i j  , = ig n−2 (2π )4 δ (4) P ab n l=1 ll + 1 ik n

˙

(2.2)

˙

a ˜b λm is the total momentum, n is the total where P ab = m=1 λm number of particles and the dots stand for any number of positive helicity gluons, i is the position of the negative-helicity gluon, while k and j are the positions of the gluons with helicities +s and −s respectively. The expression (2.2) reduces to the famous Parke–Taylor formula [14] when s = 1. The scattering amplitudes (2.2) can be used to extract splitting amplitudes of gluons and tensor-gluons [15]. Indeed, the collinear behavior of the tree amplitudes has the following factorized form [14,18,19]:



M ntree . . . , aλa , bλb , . . . ab 

→



 tree



 tree



Split−λ aλa , bλb × M n−1 . . . , P λ , . . . ,

(2.3)

λ=±1

λa λb where Splittree −λ (a , b ) denotes the splitting amplitude and the intermediate state P has momentum k P = ka + kb and helicity λ. Considering the amplitude (2.2) in different collinear limits one can get the full set of splitting amplitudes [15]:





Split+ a+s , b−s =



Split+ a−s , b



+

Split+s a , b

 +s −s







1−z z

 =

z

 

+

Split−s a , b

+s



(1 − z)2 1 , √ z(1 − z) a, b √

z2

1

z(1 − z) a, b

z

s +1

1

z(1 − z) a, b z − s +1

1

z(1 − z) a, b

P TG ( z) = C 2 (G )



z4

z

2s−2

z(1 − z)

1−z  2s−2  (1 − z)4 1 − z , + z(1 − z) z  2s−2 

(1 − z)4 1 1 2s−2 , P GT ( z) = C 2 (G ) + (1 − z) z(1 − z) 1 − z z(1 − z)  2s−2 

z4 1 1 . P TT ( z) = C 2 (G ) z2s−2 + (2.6) z(1 − z) z(1 − z) z The momentum conservation in the vertices clearly fulfills because these functions satisfy the relations

P TG ( z) = P TG (1 − z),

P GT ( z) = P TT (1 − z),

z < 1.

(2.7)

In the leading order the kernel P TG ( z) has a meaning of variation per unit t of the probability density of finding a tensor-gluon inside the gluon, P GT ( z) – of finding gluon inside the tensorgluon and P TT ( z) – of finding tensor-gluon inside the tensor-gluon. For completeness we shall present also quark and gluon kernels [4]:

P qq ( z) = C 2 ( R )

1 + z2

P Gq ( z) = C 2 ( R )

,

1 + (1 − z)2

1−z z

2 2 P qG ( z) = T ( R ) z + (1 − z) , 

(1 − z)4 1 z4 , P GG ( z) = C 2 (G ) + + z(1 − z) z(1 − z) z(1 − z)

,

(2.8)

−1 where C 2 (G ) = N, C 2 ( R ) = N2N , T ( R ) = 12 for the SU ( N ) groups. Notice that the tensor splitting kernels reduce to the quark–gluon splitting kernels (2.8) in the limit s → 1/2: 2

P GT ( z) → P Gq ( z),

P TG ( z) → P qG ( z)

,

(2.9)

and to the gluon–gluon splitting kernels in the limit s → 1:

1 2



P TG ( z) + P GT ( z) + P TT ( z) → P GG ( z).

(2.10)

Having in hand the new set of splitting kernels for tensorgluons (2.6) we can derive DGLAP equations [4–8], which will take into account these new emission processes.

, ,

(1 − z)−s+1 1 = √ . z(1 − z) a, b

3. Generalization of DGLAP equation

(2.4)

This set of amplitudes (2.4) describes the splitting G → TT, T → GT and T → TG. Since the collinear limits of the scattering amplitudes are responsible for parton evolution [4] we can extract from the above expressions the splitting probabilities for tensor-gluons. Indeed, the residue of the collinear pole in the square (of the factorized amplitude (2.3)) gives Altarelli–Parisi splitting probability P ( z):

   Split−h aha , bhb 2 sab , P ( z) = C 2 (G ) P h P ,ha ,hb

where sab = 2ka · kb = a, b[a, b]. The invariant operator C 2 for the representation R of the Lie algebra G is defined by the equation t a t a = C 2 ( R )1 and tr(t a t b ) = T ( R )δab . Substituting the splitting amplitudes (2.4) into (2.5) we will get

P TT ( z) → P qq ( z),

(1 − z ) s + 1 1 =√ , z(1 − z) a, b

Split−s a+s , b+ = √



 s −1

1−z

Split+s a−s , b+ = √



 s −1

151

(2.5)

If one supposes that there are additional partons – tensorsgluons then one should introduce the corresponding density T (x, t ) of tensor-gluons (summed over colors) in the P ∞ frame. We can derive the integro-differential equations that describe the Q 2 dependence of parton densities in this general case. They are:

dqi (x, t ) dt

α (t ) = 2π

1

 2n

f dy 

y x

  j

q ( y , t ) P qi q j

j =1

  x , + G ( y , t ) P qi G y

x

y

152

G. Savvidy / Physics Letters B 732 (2014) 150–155

dG (x, t ) dt

α (t ) = 2π

1

 2n

f dy 

y

dt

α (t ) = 2π

1

q ( y , t ) P Gq j

j =1

x

 

+ G ( y, t ) P G G dT (x, t )

  j

dy y

x

y

similar to the Altarelli–Parisi regularization [4]. We shall define them in the following way:

x

y

1

  x + T ( y, t ) P G T ,

dz

y



0

G ( y, t ) P T G

x

x

dz

y 0

 

+ T ( y, t ) P T T

x

y

(3.11)

.

The α (t ) is the running coupling constant (α = g /4π ). In the leading logarithmic approximation α (t ) is of the form 2

α = 1 + bαt , α (t )

(3.12)

where α = α (0) and b is the one-loop Callan–Symanzik coefficient, which, as we shall see below, receives an additional contribution from the tensor-gluons. Here the indices i and j run over quarks and antiquarks of all flavors. The number of quarks of a given fraction of momentum changes when a quark looses momentum by radiating a gluon, or a gluon may produce a quark–antiquark pair [4]. Similarly the number of gluons changes because a quark may radiate a gluon or because a gluon may split into a quark– antiquark pair or into two gluons or into two tensor-gluons. This last possibility is realized, because in non-Abelian tensor gauge theories there is a triple vertex GTT of a gluon and two tensorgluons of order g [9–12]. This interaction should be taken into consideration, and we added the term T ( y , t ) P G T ( xy ) in the second equation (3.11). The density of tensor-gluons T (x, t ) changes when a gluon splits into two tensor-gluons or when a tensor-gluon radiates a gluon. This evolution is described by the last equation (3.11). In order to guarantee that the total momentum is unchanged, one should impose the following constraint:

d

1 dz z

dt

 2n f 

 q i ( z , t ) + G ( z , t ) + T ( z , t ) = 0.

(3.13)

i =1

0

dz z P qq ( z) + P Gq ( z) = 0,

dz z 2n f P qG ( z) + P GG ( z) + P TG ( z) = 0,



dz z P GT ( z) + P TT ( z) = 0.

2s−2

1 k =0 k ! z2s−1

f (k) (0) zk

,

0

1 =

z+ (1 − z)+

dz

f ( z) − (1 − z) f (0) − zf (1) z(1 − z)

0

,

(3.15)



dz z 2n f P qG ( z) + P GG ( z) + P TG ( z) + b G δ( z − 1) 0

2

11

3

6

= n f T (R) −

C 2 (G ) −

12s2 − 1 6

C 2 ( G ) + b G = 0.

(3.16)

From this result we can extract the one-loop Callan–Symanzik beta function, because the momentum sum rule (3.14) implicitly comprises unitarity, thus the one-loop effects [4]. In (3.16) we have three terms which come from quark loops:

bq = −

2n f 3

T ( R ),

(3.17)

from the gluon loop:

11 6

C 2 (G )

(3.18)

and from the tensor-gluon loop of spin s:

12s2 − 1

α (t ) =

0

1

(−1)k (k) f (1)(1 − z)k k! , (1 − z)2s−1

k =0

6

C 2 (G ),

s = 1, 2, 3, 4, . . . .

(3.19)

It is a very interesting result because at s = 1 we are rediscovering the asymptotic freedom result (3.18). For larger spins the Callan– Symanzik beta function has the same signature as for the standard gluons and “accelerates” the asymptotic freedom:

0

1

f ( z) −

dz

f ( z)

1

bT =



2s−1 z+

=

2s−2

where f ( z) is any test function which is sufficiently regular at the end points. As in the standard case [4] we should add the delta function terms into the definition of the diagonal kernels so that they will completely determine the behavior of P qq ( z), P GG ( z) and P TT ( z) kernels. The first equation in the momentum sum rule (3.14) remains unchanged because there is no tensorgluon contribution into the quark evolution. The second equation in the momentum sum rule (3.14) will take the following form (see Appendix A for details):

bG =

Expressing the derivatives in (3.13) in terms of splitting kernels (3.11) one can see that the following momentum sum rules should be fulfilled:

1

0

dz

f ( z) −

0

1

f ( z)

1 dz

=

−1 (1 − z)2s +

1

 

1

f ( z)

(3.14)

0

Before analyzing these momentum sum rules let us first inspect the behavior of the tensor-gluon kernels (2.6) at the end points z = 0, 1. As one can see, they are singular at the boundary values similarly to the case of the standard kernels (2.8). Though there is a difference here: the singularities are of higher order compared to the standard case [4]. Therefore one should define the regularization procedure for the singular factors (1 − z)−2s+1 and z−2s+1 2s+1 −2s+1 reinterpreting them as the distributions (1 − z)− and z+ , +

α 1 + bαt

(3.20)

,

where

b=

(12s2 − 1)C 2 (G ) − 4n f T ( R ) 12π

,

s = 1, 2, . . .

(3.21)

Surprisingly, a similar result based on the parametrization of the charge renormalization taken in the form b = (−1)2s ( A + Bs2 ) was conjectured by Curtright [28]. Here A represents an orbital contribution and Bs2 – the anomalous magnetic moment contribution [29–31]. The unknown coefficients A and B were found by comparing the suggested parametrization with the known results for s = 0, 1/2 and 1.

G. Savvidy / Physics Letters B 732 (2014) 150–155

H2

The third equation in the momentum sum rule (3.14) will take the following form:

=

1

where



dz z P TT ( z) + P GT ( z) + b T δ( z − 1)

2

b=−

0

= −C 2 ( G )

2s +1 

1

+ bT = 0

j

j =1

(3.22)

and we can extract the one-loop coefficient of the Callan–Symanzik beta function now for tensor-gluon of spin s, which has the form

bTT = C 2 (G )

2s +1 

1 j

j =1

s = 1, 2, 3, 4, . . . .

,

(3.23)

As one can see, at s = 1 we have

  1 1 11 = C 2 (G ) 1 + + C 2 (G ), 2

3



1 + z2 1−z

P GG ( z) = 2C 2 (G )

z

z

P TT ( z) = C 2 (G )

 1−z + + z(1 − z)

(1 − z)+

P TG ( z) = C 2 (G ) +

z2s+1 −1 (1 − z)2s +

12s2 − 1 6

+

1

2s−1 (1 − z) z+

+

 δ( z − 1) .

π

+

j =1

j

⎛ dz z

(3.26)

γ

⎞

2n f P qG ( z)

P Gq ( z)

P GG ( z)

P GT ( z) ⎠

P TG ( z)

P TT ( z)

qG 2n f n GG n TG n

γ

γ γ

0

0

⎞

0

⎟

γnGT ⎠ , γnTT

(3.27)

where the corresponding integrals for quarks and gluons are well known [1]. We have to calculate the new terms in the matrix of anomalous dimensions which are the contribution of tensorgluons. For tensors-gluons kernel P TG ( z) they are:

dz zn−1 P TG ( z)



2s +n  k=2s−1



+ C 2 (G )

(−1)k (2s + 2)! 1 k!(2s + n − k)! k − 2s + 2

2s +1  k=2s−n

(1 − z)2s+1

+

2s−1 z+

1

12s − 1 2

6

n −1

dz z



(−1)k (2s + 1)! 1 k!(2s + 1 − k)! k + 1 + n − 2s



n  2s − 1

,



P TG ( z) = C 2 (G )

2s +n  k=2s−1

0

(−1)k (2s + 2)! 1 k!(2s + n − k)! k − 2s + 2





Γ (2s + 2)Γ (n − 2s + 1) + C 2 (G ) Γ (n + 3) +

˜i

with the emission of q are taken to be zero. We also ignored the infinite “stair” of emissions of tensor-gluons by tensor-gluons generated by the TT  T  vertex [9–12]. In our second approach we shall use the effective Lagrangian technique for the description of quantum corrections. For that purpose we shall generalize the results obtained for the effective Lagrangian in Yang–Mills theory long ago [29–34] to the higher spin tensor-gluons. With the spectrum of the tensor-gluons in the external chromomagnetic field λ = (2n + 1 + 2s) g H + k2 one can perform a summation of the modes and get an exact result for the one-loop effective Lagrangian similar to [29]:

(3.25)

,

2

12π

0 qq n Gq n

⎜ =⎝γ

δ( z − 1) ,

Thus we completed the definition of the kernels appearing in the evolution equations (3.11). In (3.11) we ignored the contribution of the high-spin fermions q˜ i of spin s + 1/2 [9–12], supposing that they are very heavy, thus all kernels P qq˜ , P G q˜ , P T q˜ , P q˜ q˜ , P q˜ q , P q˜ G , P q˜ T and P T q



P qq ( z)

0

⎛



(3.24)

μ2

1

−

  2s + 1 12s2 − 1 = ζ −1, C 2 (G ),

n −1 ⎝

0

1

gH

2

= C 2 (G ) 2s +1 

b ln

and ζ (−1, q) = − 12 (q2 − q + 16 ) is the generalized zeta function.2 Because the coefficient in front of the logarithm defines the beta function [29,30], one can see that (3.26) is in agreement with the result (3.19). It is also interesting to mention that the spectrum of the open strings in the background magnetic field has a similar spectrum with the gyromagnetic ration equal to two, as it was pointed out by Bachas and Fabre in [35]. Let us also calculate the matrix of the anomalous dimensions of the twist-two operators. They are represented in the terms of moments of the kernels in (3.11):

1

(1 − z)+ z 11C 2 (G ) − 4n f T ( R ) + δ( z − 1), 6  2s+1

4π

2C 2 (G )

γn =

 3 + δ( z − 1) , 2

( g H )2

1

6

and it coincides with the one loop contribution of the gluons (3.18). The coefficient grows as ln s. This growth is slower than s2 in (3.21). The reason is that, as we shall discuss at the end of this section, the splitting of tensor-gluons into pair of tensorgluons was discarded in the derivation of the evolution equation. In this article we limit ourselves by considering only emissions which always involve the standard gluons and lower-spin tensors, ignoring infinite “stair” of transitions between tensor-gluons. In summary, for the diagonal kernels P qq ( z), P GG ( z) and P TT ( z) we got:

P qq ( z) = C 2 ( R )

+

153

12s2 − 1 6

,

n  2s,

(3.28)

for the gluons-tensors kernel P GT ( z):

1

n −1

dz z 0

2

The

1 Γ ( p)

∞ 0



Γ (2s + 2)Γ (n − 1) , P GT ( z) = C 2 (G ) Γ (n + 2s + 1)

generalized −qt dt t −1+ p 1e−e−t

zeta .

function

is

defined

as

n  2s

ζ ( p , q) =

∞

1 k=0 (k+q) p

=

154

G. Savvidy / Physics Letters B 732 (2014) 150–155

1

n −1

dz z

strength at energies of order M ∼ 4 × 104 GeV = 40 TeV, which is much smaller than the scale M ∼ 1014 GeV in the absence of the tensor-gluons contribution. The value of the weak angle [36,37] remains intact and the coupling constant at the unification scale remains of the same order.

P GT ( z)

0



n −2 

(−1)k (n − 2)! 1 k!(n − k − 2)! k − 2s + 2 k=2s−1  Γ (2s + 2)Γ (n − 1) + , n  2s + 1 Γ (n + 2s + 1)

= C 2 (G )

Acknowledgements

(3.29)

and for the tensors–tensors kernel P TT ( z):

1

n +2s

dz zn−1 P TT ( z) = −C 2 (G )

1

j =2+2s

0



1

n −1

dz z

P TT ( z) = −C 2 (G )

,

j

n +2s

1

j =2+2s

0

j

n  2s,

+

n −2s j =1

1

Appendix A



j

,

n  2s + 1 (3.30)

where s = 2, 3, 4, . . . . For the nonsinglet piece the relevant anomaqq lous dimension is γn , and it does not have any additional contribution from tensor-gluons, therefore for nonsinglet pieces one can NS (n) get C N S [ αα(t ) ] An , where

A nNS =

γnqq 3C 2 ( R ) =− 2 2π b (11 + 12s − 1)C 2 (G ) − 4n f T ( R ) 

× 1−

2 n(n + 1)

+4

n  1 j =2

j

(3.31)

0

0 1 C (G ) 2s+2 2 1 − 2s+2 C 2 (G )

0

⎞

⎟ ⎠,

(3.32)

1

=

1

αi (μ)

+ 2bi ln

M

μ

,

i = 1, 2, 3,

58C 2 (G ) − 4n f T ( R ) 6π

(3.34)

.

For the SU (3)c × SU (2) L × U (1) group with its coupling constants α3 , α2 and α1 and six quarks n f = 6 and SU (5) unification group we will get 2b3 = 21π 54, 2b2 = 21π 104 , 2b1 = − 21π 4, so that 3 solving the system of Eq. (3.33) one can get

ln

M

μ

=

π 58



1

αel (μ)

−

8

1

3 αs (μ)



,

z2s+2 −1 (1 − z)2s +



= C 2 (G )

dz

+

+

(1 − z)2s+1



2s−2 z+

(−1)k (2s + 2)! (1 − z)k k!(2s + 2 − k)! (1 − z)2s−1

(−1)k (2s + 1)! zk k!(2s + 1 − k)! z2s−2

k=2s−2



2s +2  k=2s−1

2s +1 



2s +2 

(−1)k (2s + 2)! 1 k!(2s + 2 − k)! k − 2s + 2 k=2s−1  2s +1  (−1)k (2s + 1)! 1 + k!(2s + 1 − k)! k − 2s + 3

= C 2 (G )

=−

12s2 − 1 6

(3.35)

where αel (μ) and αs (μ) are the electromagnetic and strong coupling constants at scale μ. If one takes αel ( M Z ) = 1/128 and αs ( M Z ) = 1/10 one can get that coupling constants have equal

C 2 (G ).

(A.1)

The same integral we shall calculate now using the substitution w = 1 − z in the second term of the integrand and evaluating the sum

(3.33)

where we shall consider only the contribution of the lower s = 2 tensor-bosons:

2b =

dz

k=2s−2

and for arbitrary n it is given by (3.28), (3.29) and (3.30). It is interesting to know how the contribution of tensor-gluons changes the high energy behavior of the coupling constants of the Standard Model [36,37]. The coupling constants are evolving in accordance with the formulae

αi ( M )

C 2 (G )

0

A nS = γn /2π b are the eigenvalues of the matrix (3.27). For n = 2 the matrix of anomalous dimensions has the form

− 86 C 2 ( R ) 46 n f T ( R ) 8 4 γ2 = ⎜ ⎝ 6 C2(R) − 6 n f T (R)



1

1



.

The momentum sum rule integrals (3.14), (3.16) and (3.22) can be calculated in two different ways: with direct regularization of the integrals near z = 1 and near z = 0 as it is defined in (3.15), or using the substitution w = 1 − z in the second term of the integrand and then regularizing the resulting integrand near z = 1. As one can convinced both methods give the identical results. Therefore we shell first calculate the integral over P TG ( z) in (3.16) by direct regularization as follows:

0

For the singlet pieces one should take into account the mixing of S (n) gluon, fermion and tensor operators, so that C S [ αα(t ) ] An , where

⎛

This work was supported in part by the General Secretariat for Research and Technology of Greece and the European Regional Development Fund of the European Union under the contract MIS-448332-ORASY (NSRF 2007-13 ACTION, KRIPIS).



1 = C 2 (G )

dz 0

z2s+2

(1 − z)2s−1

1 = C 2 (G )

dz 0

+

z2s+1



(1 − z)2s−2

z2s+1

(1 − z)2s−1

and then one should regularize the integral near z = 1

1 C 2 (G )

dz 0

z2s+1 −1 (1 − z)2s +

1 = C 2 (G )

dz 0

2s +1 



k=2s−1

(−1)k (2s + 1)! (1 − z)k k!(2s + 1 − k)! (1 − z)2s−1



G. Savvidy / Physics Letters B 732 (2014) 150–155

the three terms are k = 2s − 1, 2s, 2s + 1

1



dz −(2s + 1)s + (2s + 1)(1 − z) − (1 − z)2

C 2 (G )

0

=−

12s2 − 1 6

C 2 (G ),

(A.2)

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