Masujima M. Applied Mathematical Methods in Theoretical Physics

Подождите немного. Документ загружается.

488 10 Calculus of Variations: Applications

The action functional is written as



<

(2π)



− m



φ(k)

φ(−k)



+ I

int

where I

int

[

φ]isgivenby

int

=−



<

(2π)

+ k

)

φ(k

)

φ(k

)

φ(k

)

φ(k

The generating functional of Green’s function is given by

Z [J] = N



exp







φ(k), ik

φ(k)



J(k)

φ(k) +‘‘iε-piece’’



Here we have



φ] =

<



φ(k)D

∗

(k).

With the scale transformations

p =



, ϕ(p) = 

φ(k),

m = 

−1

the generating functional of Green’s function is now given by

Z [J] = N



D[ϕ]exp





(2π)

L(ϕ, ip

ϕ) +Jϕ +



iε-piece





with



D[ϕ] =



Dϕ(p)Dϕ

∗

(p).

The action functional is now given by

I[ϕ] =



(2π)



−



ϕ(p)ϕ(−p)



+ I

int

[ϕ],

with the interaction term given by

int

[ϕ] =−



(2π)

+ p

)ϕ(p

10.15 Renormalization Group Equations 489

We now separate the ﬁeld variable into the slowly varying component and the

rapidly varying component as

ϕ(p) = ϕ

slow

(p) +ϕ

rapid

(p)

with

slow

(p) = 0 unless



< 1b,

rapid

(p) = 0 unless (1b) ≤



≤ 1,

b > 1.

The generating functional of Green’s function is rewritten as

Z [J] = N



[

slow

]

)

rapid

exp

)

iI[ϕ

slow

(p) +ϕ

rapid

(p)] +‘‘iε-piece’’

and we perform the path integration with respect to ϕ

rapid

. We deﬁne the new

action functional

I[ϕ

slow

], which depends only on ϕ

slow

exp

I[ϕ

slow

]



)

rapid

exp

)

slow

(p) +ϕ

rapid

(p)

We expand the new action functional

I[ϕ

slow

]intheform

I[ϕ

slow

] =



<1b

(2π)



−r



slow

(p)ϕ

slow

(−p) + I

int

[ϕ

slow

which deﬁnes z, r

, and the new parameters in I

int

[ϕ

slow

]. We set z = b

−γ

,where

we call γ the anomalous mass dimension. The generating functional of Green’s

function is now written as

Z [J] = N



D[ϕ

slow

]exp



I[ϕ

slow

] +‘‘iε-piece’’



We rescale the cutoff back to 1 by deﬁning



= bp, ϕ



) = b

−1−γ 2

slow



b).

The action functional can be written as



[ϕ



] ≡

I[ϕ

slow

] =



(2π)



− r







(p)ϕ



(−p) + I



int

)



where r



= b

2+γ

490 10 Calculus of Variations: Applications

We compute the N-point Green’s function in the momentum space as

, ..., p

;, g

) =

δJ(p

)

···

δJ(p

)

Z[J]



φ]

φ(p

) ···

φ(p

)

exp

iI[

φ] +‘‘iε-piece’’



φ]exp[iI[

φ] +‘‘iε-piece’’]

Here



<

and g

represents the bare coupling constant.

In terms of the dimensionless ﬁeld,

ϕ(p) = 

φ(p)

and the dimensionless bare coupling constant g

,wecanwrite

, ..., p

;, g

) = 

−N





, ...,





where

, ..., p

) =



D[ϕ]

ϕ(p

) ···ϕ(p

)

exp

[

iI[ϕ] + ‘‘iε-piece’’

]



D[ϕ]exp

[

iI[ϕ] + ‘‘iε-piece’’

]

We are interested only in the N-point Green’s function of the slowly varying ﬁelds

with



< 1b.

Setting ϕ(p

) = ϕ

slow

), we have

, ..., p

)



[

slow

]



D[ϕ

rapid

]

ϕ(p

) ···ϕ(p

)

exp

)

iI[ϕ

slow

+ ϕ

rapid

] + ‘‘iε-piece’’



D[ϕ

slow

]



D[ϕ

rapid

]exp

)

iI[ϕ

slow

+ ϕ

rapid

] +‘‘iε-piece’’



D[ϕ

slow

]

slow

) ···ϕ

slow

)

exp

I[ϕ

slow

] + ‘‘iε-piece’’



D[ϕ

slow

]exp

I[ϕ

slow

] +‘‘iε-piece’’

10.15 Renormalization Group Equations 491

= z

−N2



D[ϕ



]



(bp

) ···ϕ



(bp

)

exp

)



[ϕ



] +‘‘iε-piece’’



D[ϕ



]exp

)



[ϕ



] +‘‘iε-piece’’

= z

−N2

(bp

, ..., bp



To take contact with the perturbative renormalization, we set

b = µ,

where µ is the normalization point. Multiplying the above equation by 

−N

(bµ)

−N

,weobtain

, ..., p

;, g

) = z

−N2

−N



, ...,





In the perturbative renormalization, the above equation is usually written as

, ..., p

;, g

) =







, g



−N2



, ..., p

;µ, g),

where g is the renormalized coupling constant and G



, ..., p

;µ, g) is the renor-

malized N-point Green’s function. We should observe that the cutoff dependence

is isolated in the factor z

(µ, g

Renormalization Group of Gell–Mann–Low

We brieﬂy outline the Gell–Mann–Low approach to the renormalization

group. We consider the quartic self-interacting neutral scalar ﬁeld theory as a

model,



φ(x), ∂

φ(x)



=−

∂

φ(x)∂

φ(x) −

(x) −

(x).

In order to ﬁx the normalization of Green’s functions, we give a set of pre-

scriptions. We introduce a parameter µ of the dimension of mass and properly

normalize the two-point Green’s function at p

= µ

. The coupling constant g(µ)

is introduced as the value of the one-particle-irreducible (1PI) four-point Green’s

function for p

= (µ

3)(4δ

− 1), i, j = 1, 2, 3, 4. The 1PI Green’s functions are

deﬁned as those Green’s functions which cannot be made disjoint by cutting one

internal line.

The N-point Green’s function is denoted by

(N)



;µ, g(µ)



, i = 1, 2, ..., N. (10.15.1)

492 10 Calculus of Variations: Applications

When the parameter µ is equal to the physical mass m

,wehavethestandard

mass-shell renormalization and we deﬁne

= g(m

). (10.15.2)

The scale transformation argument yields the following relationship:

(N)

, g

) = Z(µ)

−N2

(N)



;µ, g(µ)



. (10.15.3)

The renormalization group is deﬁned as the group of scale transformations

of the parameter µ. Equation (10.15.3) shows the characteristic feature of the

renormalization group that the change of the scale of µ induces the change of

the coupling constant and the change of the overall normalization of Green’s

functions.

In order to express the scale change of µ, we introduce a new parameter ρ by

ρ = ln





, (10.15.4)

and rewrite Eq. (10.15.3) as

(N)

, g

) = Z(ρ)

−N2

(N)



;µ(ρ), g(ρ)



. (10.15.5)

The left-hand side of Eq. (10.15.5) is independent of ρ so that the derivative of the

right-hand side of Eq. (10.15.5) with respect to ρ must vanish. As a function of ρ,

we observe that µ(ρ), g(ρ), and Z(ρ) satisfy the boundary conditions

µ(0) = m

, g(0) = g

, Z(0) = 1. (10.15.6)

Now g(ρ) is a function of ρ and g

, so that its derivative with respect to ρ is also a

function of ρ and g

.Sinceg

canbeexpressedintermsofρ and g, ∂g∂ρ can be

regarded as a function of µm

(= exp[ρ]) and g,

∂

∂ρ

g = β





. (10.15.7)

Similarly, we have

∂

∂ρ

ln Z(ρ)

−1

= γ





. (10.15.8)

The vanishing of the derivative of Eq. (10.15.5) with respect to ρ is expressed as



∂

∂µ

+ β





∂

∂g

Nγ





(N)



;µ, g



= 0. (10.15.9)

10.15 Renormalization Group Equations 493

We deﬁne a differential operator D by

D = µ

∂

∂µ

+β





∂

∂g

. (10.15.10)

With δρ as the inﬁnitesimal parameter of the renormalization group, the inﬁnites-

imal change of Q, a function of µ and g, under the renormalization group, is given

δQ = (DQ)δρ . (10.15.11)

Renormalization Group of Callan–Symanzik

In the Gell–Mann–Low equation brieﬂy discussed above, the physical mass and the

physical coupling constant, m

and g

, are held ﬁxed and only the normalization

point µ is varied. On the contrary, both the physical mass and the physical

coupling constant are varied in the Callan–Symanzik equation. For the reason to

be discussed later, we simply denote them by m and g, respectively.

The Callan–Symanzik equationfor the N-point Green’s function is given by



∂

∂m

+β(g)

∂

∂g

Nγ

(g)



(N)

;m, g) = G

(N)

;m , g), (10.15.12)

where the right-hand side is the mass-insertion term, and for a model theory under

consideration, it is given by

G

(N)

;m, g) =



−δ(g)m

∂

∂m



(N)

;m, g),

β(g) = dg(ρ)dρ, γ

(g) = d ln Z

−1

(ρ)dρ, δ(g) = dm(ρ)dρ.

Callan–Symanzik equationsfor the j-fold mass insertion terms in the

(x)theory

are given by



∂

∂m

+β(g)

∂

∂g

Nγ

(g) − j



2 −γ

(g)







(N)

;m, g)

= 

j+1

(N)

;m , g). (10.15.13)

Here

(x) = Z

(x), Zm

∂m

= m, γ

(g) = Zm

∂ ln Z

∂m

We call Eq. (10.15.13) the generalized Callan–Symanzik equation. In order to cast

the generalized Callan–Symanzik equation, (10.15.13), in a homogeneous form,

494 10 Calculus of Variations: Applications

we introduce the generating function of the j-fold mass-insertion terms by

(N)

;m, g, K) =

∞



j=0



(N)

;m , g). (10.15.14)

This generating function satisﬁes a homogeneous equation



∂

∂m

+ β(g)

∂

∂g

Nγ

(g) −

1 +(1 −γ

(g))K

∂

∂K



(N)

;m, g, K) = 0.

(10.15.15)

Although this differential equation is homogeneous, we have introduced three

parameters, m, g,andK, to be compared with the two parameters, µ and g,inthe

Gell–Mann–Low equation, (10.15.9). Next we deﬁne the differential operator

D by

D = m

∂

∂m

+ β(g)

∂

∂g

−

(2 +δ(g))K + 1

∂

∂K

. (10.15.16)

The inﬁnitesimal change of Q, a function of m, g and K,underthe

Callan–Symanzik type renormalization group is given by

δQ = (

DQ)δρ . (10.15.17)

For example, we have

δm = mδρ , δg = β(g)δρ, δK =−

1

2 +δ(g)



K + 1

δρ . (10.15.18)

The solutions of Eq. (10.15.18) with the initial conditions,

g(0) = g,

K(0) = K, (10.15.19)

are denoted by m exp[ρ],

g(ρ), and

K(ρ).

The equation corresponding to Eq. (10.15.3) in the present approach is given by

(N)

;m, g, K) = exp







g(ρ



)



dρ





(N)



;m exp[ρ],

g(ρ),

K(ρ)



(10.15.20)

We shall choose a special value of ρ, denoted by ˜ρ,suchthat

K( ˜ρ) = 0. (10.15.21)

Then this ˜ρ is a function of g and K, and the two-point Green’s function for this

choice of ρ is given by G

(2)

;m exp[ ˜ρ],

g( ˜ρ), 0), which has a pole at

= m

exp[2 ˜ρ]. (10.15.22)

10.15 Renormalization Group Equations 495

The single particle mass, m exp[ ˜ρ], is a function of m, g,andK. From now on, we

shall consider that

= m exp[ ˜ρ] (10.15.23)

denotes the ﬁxed physical mass, and m denotes a movable normalization point. In

fact, we have

= 0, (10.15.24)

which is a consequence of the fact that m

isthepoleofthetwo-pointGreen’s

function , G

(2)

;m exp[ ˜ρ],

g( ˜ρ), 0). Substituting Eq. (10.15.23) into Eq. (10.15.24),

we ﬁnd

D ˜ρ =−1. (10.15.25)

The function ˜ρ is alternatively ﬁxed by Eq. (10.15.25) with the boundary condition,

˜ρ = 0forK = 0. Integrating the second equation of Eq. (10.15.18), δg = β(g)δρ ,

with the initial condition

g(0) = g,

K(0) = K, (10.15.19)

we obtain

˜ρ =



g( ˜ρ)

β(x)

. (10.15.26)

Applying

D on the above, and using Eqs. (10.15.25), (10.15.17), and (10.15.18), we

obtain

−1 =

D ˜ρ =

β(

g( ˜ρ))

g( ˜ρ) −

β(g)

Dg =

β(

g( ˜ρ))

g( ˜ρ) − 1.

Hence we have

g( ˜ρ) = 0. (10.15.27)

Namely, the on-shell coupling constant, g

g( ˜ρ), is also an invariant of the

Callan–Symanzik type renormalization group.

Next we shall regard K as a function of g and ˜ρ = ln[m

m], or as a function of g

and mm

. Then Green’s functions can be expressed in terms of the parameters,

, m, g,andm

,insteadofp

, m, g,andK.Wewrite

(N)



;m , g, K(g,

)



≡ G

(N)

;m, g, m

). (10.15.28)

496 10 Calculus of Variations: Applications

Taking the following equation into consideration:

= 0, (10.15.24)

we have the following equation from Eq. (10.15.15):

0 =



D +

Nγ

(g)



(N)

;m , g, m

)









∂

∂m



g,m







∂

∂g



m,m







∂

∂m



g,m

Nγ

(g)



× G

(N)

;m , g, m

)



∂

∂m

+β(g)

∂

∂g

Nγ

(g)



(N)

;m, g, m

We have a homogeneous Callan–Symanzik equation involving m and g alone,



∂

∂m

+ β(g)

∂

∂g

Nγ

(g)



(N)

;m , g, m

) = 0. (10.15.29)

Lastly we give K as a function of g and ˜ρ. From Eqs. (10.15.18), (10.15.19), and

(10.15.21), we immediately obtain

K =



˜ρ

dρ exp





dρ



2 +δ



g(ρ)





, (10.15.30)

where

g(ρ) is deﬁned by the following equation:

ρ =



g(ρ)



β(g



)

. (10.15.31)

Since Green’s functions in the two alternate approaches depend on the two

parameters differently, we distinguish them by subscripts, ‘‘GML’’ and ‘‘CS.’’ The

renormalization group equations are given by



∂

∂m

+ β





∂

∂g

Nγ





(N)

GML

;m , g) = 0, (10.15.9)



∂

∂m

+ β(g)

∂

∂g

Nγ

(g)



(N)

;m , g) = 0. (10.15.29)

We assume that these two Green’s functions are related to each other as

(N)

GML

;m, g) = Z

N2

(N)



;m , Z

−1



. (10.15.32)

Here Z

and Z

are the functions of g and mm

,andwhenm = m

,bothsides

of Eq. (10.15.32) become Green’s functions renormalized on the mass shell. Thus,

= Z

= 1form = m

. (10.15.33)

10.15 Renormalization Group Equations 497

We shall deﬁne



= Z

−1

g, (10.15.34)

D = m



∂

∂m



+ β





∂

∂g



, (10.15.35)



= m



∂

∂m





+ β(g



)



∂

∂g





. (10.15.36)

In order to study the transformation (10.15.34), we assume

D =



. (10.15.37)

We note that

∂

∂g





∂g



∂g



−1

∂

∂g



∂

∂m





= m



∂

∂m



− m



∂g



∂m





∂g



∂g



−1

∂

∂g

Thus we have



= m



∂

∂m





∂g



∂g



−1



−m

∂g



∂m

+β(g



)



∂

∂g

and by the assumption (10.15.37), we obtain



∂

∂m

+ β





∂

∂g





= β(g



), (10.15.38)



∂

∂m

+ β





∂

∂g



F(g



) = 1withF(g



) =





β(x)

. (10.15.39)

Equation (10.15.39) determines g



as a function of g and mm

only. The boundary

condition is given by



= g for m = m

, (10.15.40)

which follows from Eqs. (10.15.33) and (10.15.34).

Lastly, by Eqs. (10.15.9), (10.15.29), (10.15.32), and (10.15.37), we obtain the

equation determining Z



∂

∂m

+ β





∂

∂g

−γ



) + γ





= 0. (10.15.41)

Once g



= Z

−1

g and Z

are known, the relationship between the two sets of

Green’s functions is completely determined by Eq. (10.15.32).