NJL fit

(1)

Parameter fixing in the two-flavor NJL model

Two-flavor NJL model in the vacuum

We consider the simplest type of the NJL model for two quark flavors, defined by the Lagrangian

L= ¯ψ(i/∂−m)ψ+G( ¯ψψ)2+ ( ¯ψiγ5~τ ψ)2

. (1)

This Lagrangian has the expected _SU(2)L×SU(2)R×U(1)V chiral symmetry and breaks the

axial_U(1)A. In any kind of regularization scheme to be considered in the following, the model

is characterized by three parameters: the coupling G, the cutoff parameter Λ, and the current quark mass m. These are adjusted in order to reproduce the chiral condensate, pion mass and decay constant in the vacuum.

The starting point is to introduce auxiliary fields in the sigma and pion channels by adding to the Lagrangian the term ∆L = − 1

4G(σ + 2Gψψ¯ )

2 ₋ 1

4G(~π+ 2Gψ¯iγ5~τ ψ)

2_{, upon which the}

Lagrangian becomes

L=−σ

2₊_~_π2

4G + ¯ψ(i∂/−M −iγ5~τ ·~π)ψ, (2)

where M = m+σ denotes the constituent quark mass. Integrating now out the quarks leads to the effective action

Γeff[σ, ~π] =−

1 4G

Z

d4x(σ2+~π2)−i Tr log(i∂/−M −iγ5~τ ·~π

| {z }

S−1

). (3)

First of all, we derive the general form of the gap equation. Setting~π =~0 (no pion condensate) and σ to a constant, the effective action becomes Γeff[σ] =−Veff(σ)Ω, where Ω is the volume of

spacetime. The effective potential is given by

Veff(σ) =

σ2

4G +

i

ΩTr log(i∂/−M). From here the gap equation follows,

σ

2G =

i

ΩTr(i/∂−M)

−1 _{= 4i}_N

cNfM

Z _d4_k

(2π)4

1

k2₋_M2. (4)

(The factor 4 comes from the trace over Dirac space, while Nc and Nf arise from the trivial

traces over color and flavor indices.)

In order to determine the physical pion mass, we need to calculate its inverse propagator. In coordinate representation, this is defined as

χab(x, y) =

δ2Γeff

δπa(x)δπb(y)

(2)

Thanks to the fact that the quark Dirac operator S−1 is linear in the background pion fields, this results in the momentum-space expression

χab(p) =−

δab

2G+ i

Z

d4k

(2π)4 Tr

∂S−1 ∂πa

S(k+p)∂S

−1

∂πb

S(k)

=

=−δab

2G + i

Z

d4_k

(2π)4 Tr

(−iγ5τa)

1

/

k+_/p−M(−iγ5τb)

1

/ k−M

=

=−δab

2G + 4iNcNfδab

Z _d4_k

(2π)4

k·(k+p)−M2

[(k+p)2₋_M2_](_k2₋_M2₎. (5)

We define χab(p) = δabχ(p). The inverse propagator χ(p) can be expressed as a function of p2

alone by the following trick. Writing

k·(k+p)−M2

[(k+p)2 ₋_M2_](_k2₋_M2₎ =

1 2

[(k+p)2₋_M2_{] + (}_k2₋_M2₎₋_p2

[(k+p)2₋_M2_](_k2₋_M2₎ =

= 1 2

1

k2₋_M2 +

1

(k+p)2₋_M2 −

p2

[(k+p)2₋_M2_](_k2₋_M2₎

,

the first two terms are identical under the integral upon a shift of the integration variable.1 Using then the gap equation (4), we obtain

χ(p2) =− 1

2G

1− σ

M

+ 2NcNfp2I(p2), I(p2) =−i

Z _d4_k

(2π)4

1

[(k+p)2₋_M2_](_k2₋_M2₎.

(6) In this form the existence of a massless pole in the chiral limit is manifest.

In order to determine the pion decay constant, we need to know the coupling, gπqq, of the

pion state |~π(p)i to the interpolating field, ¯ψiγ5~τ ψ, which enters the NJL Lagrangian. This is

extracted from the residuum of the pion propagator at the pole as

1

g2

πqq

= dχ(p

2₎

dp2

p2₌_M2

π

. (7)

Carrying out the derivative and using the pion pole condition,χ(M2

π) = 0, we find

1

g2

πqq

= m

2GM M2

π

+ 2NcNfMπ2I

0

(M_π2). (8)

The pion decay constant itself is determined from the coupling of the pion to the axial current,

h0|ψγ¯ µγ5

τa

2ψ|πb(p)i= ip

µ_f πδab.

The coupling of the axial current to the pion interpolating field can be found by adding an external axial-vector field, i.e., replacing

S−1 →S_A−1 = iγµ∂µ−iγ5

~ τ

2 ·

~ Aµ

−M−iγ5~τ ·~π,

(3)

and using the effective action as in Eq. (5),

δ2_Γ eff

δπaδAbµ

(p) = i

Z _d4_k

(2π)4Tr

∂S_A−1 ∂πa

S(k+p)∂S

−1

A

∂Abµ

S(k)

=

= i

Z

d4_k

(2π)4 Tr

(−iγ5τa)

1

/

k+_/p−Mγ

µ_γ

5

τb

2 1

/ k−M

= 2iNcNfM δabpµI(p2).

In order to get the coupling of the axial current to the pion state, we have to augment this amplitude with the above determined gπqq, hence

fπ

gπqq

= 2NcNfM I(Mπ2) =

m

2GM2

π

, (9)

where we used the pion pole condition (6) in the last step. This relation is exact in the mean-field approximation and independent of the choice of the regulator.

Equations (4), (6), (8), and (9), together with the relation

huu¯ i=hdd¯ i=− σ

4G,

which follows from the definition of the auxiliary field σ, constitute a set of equations which determine the values of the model parameters in terms of the observable quantities. In the following, we will evaluate their specific form within different regularization schemes.

3-dimensional cutoff

In this scheme the integration over frequencies is not affected and can be carried out analytically. One is then left with a three-momentum integral which is regulated using a sharp cutoff. First of all, the gap equation (4) becomes

σ

2G = 4NcNfM

Z Λ _d3_k

(2π)3

Z _d_ω

2π

1

ω2₊_k2₊_M2 = 4NcNfM

Z Λ _d3_k

(2π)3

1 2k

,

wherek =

√

k2₊_M2_{. The integral}_I₍_p2_{) is evaluated by conveniently setting}_pµ _{= (iΩ}_,₀_{) and}

using residuum theorem,

Z Λ

d3_k

(2π)3

Z

dω

2π

1 [(ω+ Ω)2₊2

k](ω2+2k)

=

Z Λ

d3_k

(2π)3

1 2k

1

(ik−Ω)2+2k

+ 1

(ik+ Ω)2+2k

=

Z Λ _d3_k

(2π)3

1 2kΩ

1 Ω−2ik

+ 1

Ω + 2ik

=

Z Λ _d3_k

(2π)3

1

k(42_k+ Ω2) .

By analytic continuation back to Minkowski space we find

I3D(p2) =

Z Λ _d3_k

(2π)3

1

(4)

The pion pole condition follows straightforwardly from Eq. (6), but can further be slightly simplified using the gap equation,

1 2G =

σ

2GM + 2NcNfM

2

πI(Mπ2) = 2NcNf Z Λ

d3k

(2π)3

1

k

+ M

2

π

k(42k−Mπ2)

=

= 8NcNf Z Λ

d3k

(2π)3

k

42

k−Mπ2

= 2NcNf Z Λ

d3k

(2π)3

1 2k+Mπ

+ 1

2k−Mπ

.

The pion coupling gπqq is in this case most easily expressed using directly Eq. (6),

1

g2

πqq

= 2NcNf[I(Mπ2) +M

2

πI

0

(M_π2)] = 2NcNf

Z Λ _d3_k

(2π)3

1

k

1 42

k−Mπ2

+ M

2

π

(42

k−Mπ2)2

=

= 8NcNf Z Λ

d3k

(2π)3

k

(42

k−Mπ2)2

.

Summary of formulas for 3-dimensional cutoff

gap equation: σ= 4GNcNfM

Z Λ _d3_k

(2π)3

1

k ,

pion pole condition: 1 = 4GNcNf

Z Λ _d3_k

(2π)3

1 2k+Mπ

+ 1

2k−Mπ

,

chiral condensate: huu¯ i=− σ

4G,

pion–quark coupling: 1

g2

πqq

= 8NcNf

Z Λ _d3_k

(2π)3

k

(42

k−Mπ2)2

,

pion decay constant: fπ

gπqq

= m

2GM2

π

.

4-dimensional cutoff

In this case the manipulation of divergent integrals is most sloppy since it involves shifts in cut-off integrals. The gap equation follows immediately from Eqs. (4) and (14). To evaluate the function I(p2), we employ the Feynman parameterization and subsequently use Eq. (15),

I4D(p2) =−i

Z 1

0

dx

Z Λ _d4_k

(2π)4

1

[k2₋_M2₊_p2_x₍₁₋_x_)]2,

I4D(p2) =

1 16π2

Z 1

0

dx

log

1 + Λ

2

M2₋_p2_x₍₁₋_x₎

− Λ

2

Λ2₊_M2₋_p2_x₍₁₋_x₎

.

From the pion propagator (6) we then simply get _Mm = 4GNcNfMπ2I4D(Mπ2). The derivative of

I4D(p2) is also straightforward, denoting C =M2−p2x(1−x),

I_4D0 (p2) = 1 16π2

Z 1

0

dx

"_Λ2

C2x(1−x) 1 + Λ_C2 −

Λ2x(1−x) (Λ2₊_C₎2

#

= Λ

4

16π2

Z 1

0

dx x(1−x) C(Λ2₊_C₎2.

(5)

Summary of formulas for 4-dimensional cutoff

gap equation: σ= G

2π2NcNfM

Λ2−M2log

1 + Λ

2

M2

,

pion pole condition: m

M = 4GNcNfM

2

πI4D(Mπ2),

4G,

g2

πqq

= m

2GM M2

π

+ NcNf 8π2 M

2

πΛ

4

Z 1

0

dx x(1−x) C(Λ2 ₊_C₎2,

gπqq

= m

2GM2

π

,

C =M2−M_π2x(1−x).

Pauli–Villars regularization

The Pauli–Villars regularization is technically slightly more complicated than the sharp mo-mentum cutoffs discussed previously. The reward for the effort is that the regularization can be defined consistently already at the level of effective action and all physical quantities subse-quently derived in an unambiguous way. Also, the integrations are not constrained so that the integration variables can be freely shifted, and all results are manifestly covariant.

We will work in Euclidean space to ensure positivity of the spectrum where needed. The naive Euclidean action, analogous to Eq. (3), reads

ΓE_eff[σ, ~π] = 1 4G

Z

dτd3x(σ2+~π2)−Tr logD.

The limits of integration over imaginary time are not specified, allowing for generalization to nonzero temperature. The Dirac operator D reads

D=−γ0 ∂

∂τ + i~γ· ~

∇ −M −iγ5~τ·~π.

Introducing a set of Hermitian Euclideanγ-matrices byγE

0 =−γ0,~γE = i~γ, so that they satisfy

{γE

µ, γνE}= 2δµν, the Dirac operator becomes

D=γ_µE∂µ−M −iγ5τ~·~π ≡γ0E(∂τ +H).

The last equation defines the Hermitian Hamiltonian as H = −iα~ ·∇~ +γ0M + iγ0γ5~τ ·~π. In

the following it will be important that

D†D=−∂_τ2+H2₋_[_∂

τ,H].

We are now ready to define the effective action in the Pauli–Villars regularization,

ΓE_eff[σ, ~π] = 1 4G

Z

dτd3x(σ2+~π2)− 1

2

X

j

(6)

The coefficients cj are chosen in order to cancel the divergences in the effective action. Here

we will consider explicitly two distinct cases:

• Two subtractions: c0,1,2 = 1,−2,1. In this case the action as defined by Eq. (10) still has

a logarithmic divergence; this can be removed by subtracting the value of the action at a conveniently chosen reference point. Also, all quantities derived by differentiation of this action are automatically finite.

• Three subtractions: c0,1,2,3 = 1,−3,3,−1. Here the action is rendered completely finite

by the subtractions.

In the following we will often abbreviate Ξj = D†D +jΛ2 and Mj2 = M2 +jΛ2. All

four-dimensional integrals will be understood in the Euclidean space without indicating the index

E _explicitly.

The presence of the square D†_D _{in the action implies that we cannot use our naive formulas}

(4), (6), (8), and (9), and have to derive them consistently starting from Eq. (10). The gap equation follows as

σ

2G =

1 2Ω

X

j

cjTr 2MΞ−j1

= 4NcNfM X

j

cj Z

d4_k

(2π)4

1

k2₊_M2

j

.

We used the fact that at ~π =~0, Ξj = −∂τ2+H2 +jΛ2 → k2 +Mj2 in the momentum

repre-sentation. To evaluate the integral we use the Wick-rotated Eq. (14) with an artificial cutoff. After the summation overj, this cutoff is sent to infinity. We thus obtain

σ = G

2π2NcNfM

X

j

cjMj2logMj2 =

= G

2π2NcNfM

M2log

1− Λ

4

(M2_{+ Λ}2₎2

+ 2Λ2log

1 + Λ

2

M2_{+ Λ}2

,

= G

2π2NcNfM

M2

log M

2

M2_{+ 3Λ}2 + 3 log

1 + Λ

2

M2 _{+ Λ}2

−3Λ2log

1− Λ

4

(M2_{+ 2Λ}2₎2

,

for the cases with two and three subtractions, respectively.

To derive the pion propagator, we functionally differentiate the action,

δΓE

eff

δπa(x)

= πa(x)

2G −

1 2

X

j

cjTr

δD†

δπa(x)

D+D† δD

δπa(x)

Ξ−_j1

= πa(x) 2G −Re

X

j

cjTr

δD†

δπa(x) DΞ−_j1

,

δ2_ΓE

eff

δπa(x)δπb(y)

= δab

2Gδ(x−y) + Re

X

j

cjTr

− δD

†

δπa(x)

δD

δπb(y)

Ξ−_j1+ δD

†

δπa(x)

DΞ−_j1δ(D

†_D₎

δπb(y)

Ξ−_j1

.

Taking into account that once the differentiation is performed, the background fields σ, ~π are set constant, we can readily go into momentum space to obtain the Euclidean inverse pion propagator,

χE_ab(p) = δab 2G+ Re

X

j

cj

Z _d4_k

(2π)4 Tr

−∂D

†

∂πa

∂D

∂πb

Ξ−_j,k1+

+∂D

†

∂πa

Dk+pΞ−j,k1+p

∂D†

∂πb

DkΞ−j,k1+

∂D†

∂πa

Dk+pΞ−j,k1+pD

†

k+p

∂D

∂πb

Ξ−_j,k1

(7)

As a byproduct of this construction we observe that the first term above, which comes from the fact that operator in the log in the action is quadratic in the background fields, is absent in Eq. (5). It turns out to be exactly the same tadpole integral as appears in the gap equation, and in the chiral limit makes the masslessness of the pion manifest. Indeed, we immediately obtain

Tr

∂D†

∂πa

∂D

∂πb

Ξ−_j,k1

= 1

k2 ₊_M2

j

Tr[(iγ5τa)(−iγ5τb)] =

4NcNfδab

k2₊_M2

j

.

This combines with the gap equation and we again find that the inverse propagator is diagonal in the isospin space,

χE(p) = 1 2G

1− σ

M

+NcNf X

j

cj Z

d4_k

(2π)4

TrD(−γ5Dk+pγ5Dk+γ5Dk+pD

†

k+pγ5)

[(k+p)2₊_M2

j](k2+Mj2)

.

In all the above formulas, Dk =−ikµγµE −M. The Dirac trace is now easily evaluated as

4[−k·(k+p)−M2+ (k+p)2+M2] = 4p·(k+p) = 2[(k+p)2+p2 −k2].

The contributions of the terms (k+p)2 and k2 cancel each other upon a shift of the integration variable in the former (which is a legal operation here!), yielding

χE(p) = 1 2G

1− σ

M

+ 2NcNfp2 X

j

cj

Z _d4_k

(2π)4

1 [(k+p)2₊_M2

j](k2+Mj2)

. (12)

This is analogous to Eq. (6). Using the Feynman parameterization and the Wick-rotated Eq. (15) we obtain the inverse propagator as a function of momentum squared,

χE(p2) = 1 2G

1− σ

M

− 1

8π2NcNfp 2X

j

cj Z 1

0

dx log[M_j2+p2x(1−x)].

As a last step, we Wick-rotate back to Minkowski space and evaluate the integral over the Feynman parameter by means of Eq. (16),

χ(p2) = m 2GM +

1

8π2NcNfp 2X

j

cj 

logM_j2+

2

p

p2

q

4M2

j −p2arctg

p

p2

q

4M2

j −p2 

.

To derive the expression for gπqq, we differentiate the function

f(x) =

q

4M_j2−x

√

x arctg

√

x

q

4M2

j −x

=yarctg1

y with y=

s

4M2

j

x −1,

f0(x) = arctg1

y +y

−1

y2 1 + _y12

!

−4M

2

j

x2

2

q

4M2

j

x −1

=− 1

2x√x

4M2

j q

4M2

j −x

arctg

√

x

q

4M2

j −x

+ 1

2x.

From here we assert [note the different sign as compared to Eq. (7); this is due to the definition of the Euclidean action]

1

g2

πqq

=− dχ(p

2₎

dp2

p2₌_M2

π

= m

2GM M2

π

+NcNf 8π2

X

j

cj

4M2

j

Mπ q

4M2

j −Mπ2

arctg_q Mπ 4M2

j −Mπ2

(8)

As a last thing we would like to verify that Eq. (9) remains valid when one starts with the action (10). Having calculated δΓEeff

δπa(x), we proceed to differentiate with respect toAbµ. In effect,

we simply have to replace πb with Abµ in Eq. (11) and drop the δab term,

δ2ΓE_eff

δπaδAµb

(p) = X

j

cj Z

d4k

(2π)4 Tr

−∂D

†

∂πa

∂D

∂Aµ_bΞ

−1

j,k +

∂D†

∂πa

Dk+pΞ−j,k1+p

∂D†

∂Aµ_b DkΞ

−1

j,k+

+∂D

†

∂πa

Dk+pΞ−j,k1+pD

†

k+p

∂D

∂Aµ_bΞ

−1

j,k

=δab

NcNf

2

X

j

cj Z

d4_k

(2π)4

TrD[(iγ5)Dk+p(γµEγ5)Dk]

[(k+p)2₊_M2

j](k2+Mj2)

.

Only the middle term survives the Dirac trace, since in the other two the Dirac structure of the combined propagators is trivial. The remaining Dirac trace reads 4iM pµ whence we find

fπ

gπqq

= 2NcNfM X

j

cj

Z _d4_k

(2π)4

1 [(k+p)2₊_M2

j](k2+Mj2)

.

This, together with Eq. (12) evaluated at p2 =−M_π2, finally yields the desired result (9).

Summary of formulas for Pauli–Villars regularization

gap equation: σ= G

2π2NcNfM

X

j

cjMj2logM

2

j,

pion pole condition:−m

M =

G

4π2NcNfM 2

π X

j

cjlogMj2+

2

Mπ

2

X

j=0

cj q

4M2

j −Mπ2arctg

Mπ q

4M2

j −Mπ2 !

,

4G,

g2

πqq

= m

2GM M2

π

+NcNf 8π2

X

j

cj

4M_j2

Mπ q

4M2

j −Mπ2

arctg_q Mπ 4M2

j −Mπ2

,

gπqq

= m

2GM2

π

.

The Pauli–Villars coefficients read c0,1,2 = 1,−2,1 and c0,1,2,3 = 1,−3,3,−1, respectively, and

M_j2 =M2 +jΛ2. Some of the sums may then be expressed explicitly as follows,

2

X

j=0

cjlogMj2 = log

1− Λ

4

(M2_{+ Λ}2₎2

,

3

X

j=0

cjlogMj2 = log

M2

M2_{+ 3Λ}2 + 3 log

1 + Λ

2

M2_{+ Λ}2

(9)

2

X

j=0

cjMj2logMj2 =M2log

1− Λ

4

(M2_{+ Λ}2₎2

+ 2Λ2log

1 + Λ

2

M2_{+ Λ}2

,

3

X

j=0

cjMj2logM

2

j =M

2

log M

2

M2_{+ 3Λ}2 + 3 log

1 + Λ

2

M2_{+ Λ}2

−3Λ2log

1− Λ

4

(M2_{+ 2Λ}2₎2

.

Meson and diquark masses

Having fixed the parameters of the NJL model, we can calculate various quantities of physical interest. Here we shall look in more detail at the propagators of the scalar and pseudoscalar mesons and the scalar diquark.

The pion propagator was determined already in Eq. (6). The sigma propagator is found using the same argument, inserting the unit matrix in place of −iγ5τa,b,

χ(p) =− 1

2G+ i

Z _d4_k

(2π)4 Tr

∂S−1

∂σ S(k+p) ∂S−1

∂σ S(k)

=

=− 1

2G+ i

Z

d4k

(2π)4 Tr

1

/

k+_/p−M

1

/ k−M

=

=− 1

2G + 4iNcNf

Z _d4_k

(2π)4

k·(k+p) +M2

[(k+p)2₋_M2_](_k2 ₋_M2₎.

Using the same trick as before,

k·(k+p) +M2

[(k+p)2₋_M2_](_k2₋_M2₎ =

1 2

1

k2₋_M2 +

1

(k+p)2₋_M2 +

4M2₋_p2

[(k+p)2 ₋_M2_](_k2₋_M2₎

,

we arrive at the formula, analogous to Eq. (6),

χσ(p2) = −

1 2G

1− σ

M

+ 2NcNf(p2−4M2)I(p2). (13)

To study the properties of the scalar diquark, we add to the Lagrangian (1) the corresponding interaction term with an adjustable coupling H,

Lqq =H ψ

C

biγ5abcijψcj

2

,

where a, b, c and i, j are the color and flavor indices, respectively. This interaction is again decoupled using the Hubbard–Stratonovich transformation, i.e., by adding to the Lagrangian the term ∆Lqq =−₄1_H

φa+ 2Hψ_biCγ5abcijψcj

2

. As a result, the semibosonized Lagrangian (2) acquires the contribution

−|φa|

2

4H −

1 2φ

∗

aψCbiγ5abcijψcj+

1

2φaψbiγ5abcijψ

C

cj.

The color and flavor structure of the diquark is specified by the matrices

(10)

The inverse diquark propagator is then given by the expression

χab(p) =−

δab

4H +

i 2

Z

d4_k

(2π)4 Tr

Pa

1

/

k+_/p−MPb

1

/ k−M

,

where the prefactor 1₂ comes from the symmetry of the quark–quark bubble. The color–flavor trace is easily found to be Trc,f(PaPb) =−2δabNf. Following now the same line of argument as

for the mesons leads to

χ(p) =− 1

4H −iNf

Z

d4k

(2π)4 TrD

γ5

1

/

k+_/p−Mγ5

1

/ k−M

=

=− 1

4H + 4iNf

Z _d4_k

(2π)4

k·(k+p)−M2

[(k+p)2₋_M2_](_k2₋_M2₎ =−

1 4H +

σ

2GNcM

+ 2Nfp2I(p2).

Let us summarize all the propagators as expressed in terms of the single loop function I(p2),

χπ(p2) =−

1 2G

1− σ

M

+ 2NcNfp2I(p2),

χσ(p2) =−

1 2G

1− σ

M

+ 2NcNf(p2−4M2)I(p2),

χ∆(p2) =−

1 4H +

σ

2GNcM

+ 2Nfp2I(p2).

Critical coupling for diquark binding

Depending on the value of the couplingH, the diquark may either be bound, or it may represent a resonance in the quark–quark spectrum, or the inverse propagatorχ∆(p2) may not have any

zero at all. Obviously the diquark ceases to be bound when the corresponding pole enters the two-particle continuum, i.e., appears atp2 = 4M2. This yields immediately the general formula

Hc=

1 4₂_GNσ

cM + 8NfM

2_I₍₄_M2₎.

However, for the sharp cutoff regularization schemes, this can be cast in a simple closed form.

Starting with the three-dimensional cutoff, we write, making use of the gap equation,

σ

2GNcM

+ 8NfM2I3D(4M2) = 2Nf

Z Λ _d3_k

(2π)3

1

k

+ 8NfM2

Z Λ _d3_k

(2π)3

1

k(42k−4M2)

=

= 2Nf

Z Λ _d3_k

(2π)3

k 2

k−M2

= 2M

2

π2

Z sinh−1 Λ_M

0

dt cosh2t= M

2

π2

t+ sinhtcoshtsinh−1 Λ_M

0 ,

where we employed the hyperbolic substitution as explained in the appendix. Finally we thus get the compact expression

Hc,3D =

π2

4 Λ√Λ2₊_M2₊_M2_sinh−1 Λ

(11)

In all regularization schemes, one has to keep in mind that inside the two-particle continuum the functionI(p2_{), and hence also the inverse propagator, acquires nonzero imaginary part. In}

case of the three-dimensional cutoff, the real part ofI(p2_{) is found by a proper principal-value}

integration over the radial momentum variable. For the covariant regulators we have already performed the momentum integration and just need to evaluate the integral over the Feynman parameter. Assuming thatp2 _<_4Λ2_{, the real part of} _I

4D can be written as

ReI4D(p2) =

1 16π2

Z 1

0

dx

log

Λ2+M2−p2x(1−x)

M2

− Λ

2

Λ2₊_M2₋_p2_x₍₁₋_x₎−

−log

1− p

2

M2x(1−x)

,

where the last integral is calculable analytically. For p2 = 4M2, it is evaluated with the help of Eq. (16) and we get

I4D(4M2) =

1 8π2 +

1 16π2

Z 1

0

dx

log

Λ2₊_M2₍₁₋₂_x₎2

M2

− Λ

2

Λ2 ₊_M2₍₁₋₂_x₎2

=

= 1

16π2

2 +

Z 1

0

dt

logΛ

2₊_M2_t2

M2 −

Λ2

Λ2₊_M2_t2

= 1

16π2

log

1 + Λ

2

M2

+ Λ

M arctg M

Λ

.

Using again the explicitly integrated gap equation, the final result for the critical coupling reads

Hc,4D =

π2

2

1

Λ2₊_M2_{log 1 +} Λ2

M2

+ 2MΛ arctgM_Λ .

Appendix: Some useful formulas

Loop integrals and their finite parts in the 4-dimensional cutoff scheme:

Z Λ _d4_p

(2π)4

1

p2 ₋_C =−

i 16π2

Λ2−Clog

1 + Λ

2

C

, (14)

Z Λ _d4_p

(2π)4

1

(p2₋_C₎2 =

i 16π2

log

1 + Λ

2

C

− Λ

2

Λ2₊_C

, (15)

Z 1

0

dx log[a2−b2x(1−x)] = 2

−1 + loga+

√

4a2₋_b2

b arctg

b

√

4a2₋_b2

, (16)

Z 1

0

dx

a2 ₋_b2_x₍₁₋_x₎ =

4

b√4a2₋_b2 arctg

b

√

4a2₋_b2. (17)

Momentum integrations with 3-dimensional cutoff: Integrals of the type

Z Λ

d3k

(2π)3f(k) =. . .

can be dealt with using the substitution k =Msinht, k=Mcosht,

. . .= M

3

2π2

Z sinh−1 Λ_M

0

(12)

Specific integrals:

Z Λ _d3_k

(2π)3

1

k

= 1

4π2

Λ√Λ2₊_M2₋_M2_sinh−1 Λ

M

=

= 1

4π2

Λ2− M

2

log 4Λ

2

M2 −1

+O

M2

Λ2

,

Z Λ

d3k

(2π)3 k =

1 16π2

Λ(2Λ2+M2)√Λ2₊_M2₋_M4_sinh−1 Λ

M

=

= 1

8π2

Λ4+ Λ2M2− M

4

log4Λ

2

M2 −

1 2

+O

M2

Λ2