Investigation of the mass renormalization method in quantum field theory

I I

7.A

]

Nuclear Physics 43 (1963) 45--56; ~ ) North-Holland Publishing Co., Amsterdam

I

N o t to be reproduced by photoprint or microfilm without written permission from the publisher

I N V E S T I G A T I O N OF

THE

MASS

RENORMALIZATION METHOD

IN Q U A N T U M F I E L D T H E O R Y M. RAM ¢ and N. ROSEN Department of Physics, Israel Institute of Technology, Haifa, Israel *

Received 20 November 1962 Abstract: A technique is described for checking the mass renormalization method in quantum field

theory in a special case. This suggests a way in which the usual mass renormalization conditions for quantum electrodynamics and other field theories can be written down immediately. 1. Introduction

As is well-known, the perturbation and renormalization theories, developed by Tomonaga, Schwinger, Feynman and Dyson, caused considerable controversy. The theories appear to account, both qualitatively and quantitatively, for the observed phenomena in quantum electrodynamics, but the mathematical treatment is not entirely satisfactory, especially as regards the infinite quantities appearing in the theory and the fact that one develops the scattering matrix in a series which probably does not converge. In this paper a method is described by means of which one is able to check the mass renormalization process in a simple case, and in this way perhaps throw some new light on the theory. From perturbation and renormalization theory one gets equations from which, at least in principle, it is possible to calculate the changes in the masses of the particles due to the interaction. I f we knew the bare masses of the particles, we could calculate the dressed masses tt and compare the result with the experimental masses. However, the bare masses of the elementary particles are unknown, since it is impossible to stop the interaction between the fields, and the mass that is measured is always the dressed mass. In this work we artificially create a situation in which the bare mass is known. We assume two real scalar fields ttt without interaction and described by a Lagrangian density in the usual way. A linear transformation of the field functions is performed, such that the new Lagrangian density is made up of a Lagrangian density representing two real scalar fields together with an interaction term. F r o m the free-field part of the new Lagrangian it is possible to determine the bare masses of the particles it t Now at the Physics Department of Columbia University, New York, N. Y. tt The dressed mass is the bare mass+the change in mass due to the interaction. ttt This is just for the sake of simplicity, as the method may also be used with spinor fields. 45

46

M. RAM AND N. ROSEN

describes, and by making use of the perturbation and mass renormalization methods o n e car~ calculate the changes in the masses of the particles due to the "interaction". Tl~ese changes,~ added to the bare masses, must be equal to the dressed masses o f the particles, which are just the masses of the particles of the non-interacting fields before the transformation. This follows from the fact that the physical content of the system must be unchanged by the transformation. Any disagreement would indicate a deficiency of the perturbation and mass renormalization theories. The notation used is as close as possible to that of Schweber 1). The " n a t u r a l " system of units is used, i.e. h = c = 1. The metric is taken so that g00 = __911 = __g22 = __033 ~- 1;

gttV = 0,

# # V.

(1.1)

Whenever a Greek index appears twice in a term, summation over this index from 0 to 3 is to be carried out. 2. Transformation of Fields

We start from the Lagrangian density 2

Se = ½ ~ . ( 9 j=l

p~v:@j,,~bj, V". - /./2j.~,j:), .t7~2

(2.1)

corresponding to two non-interacting real scalar meson fields ~1 and ~b2 associated with particles of masses gl and #z, respectively. The ~bj satisfy the usual equal-time commutation relations [qSk(x, t), n,(x', t)] = i6ktf(X--X'), [q~k(x, t), ~b,(x', t)] = [lrk(X, t), Zt(X', t)] = 0,

(2.2) k, l = 1, 2,

where zcj is the m o m e n t u m density conjugate to the field qSj, = 4,j, o .

(2.3)

W e now perform the following transformation to new field operators ~b~ and ~'2: ~bl(x, t) = (cos O)~kl(x, t ) - ( s i n O)~2(x, t), ~b2(x, t) = (sin O)Ol(x, t ) + ( c o s 0)~,2(x, t),

(2.4)

where 0 is an arbitrary parameter (0 =< 0 < ~17~). Substituting the expressions (2.4) into the Lagrangian density (2.1), one gets 2

=

(a

.,.

.

2.2

j=l

(2.5)

where m 2 = p2 cos 2 0 + #2 sin 2 0, m22 = #2 sin 2 0+#22 cos 2 0, ?

1 2 2 ~(/~z-#~) sin 20.

(2.6) (2.7) (2.8)

THE MASS RENORMALIZATION METHOD

47

It can be easily shown that the transformation (2.4) has left the canonical formalism invariant, i.e., that the correct field equations can be derived from the Lagrangian density (2.5) in the usual manner, and that the fields ~k1 and ~2 satisfy commutation relations analogous to the commutation rules (2.2). The Lagrangian density (2.5) describes two real scalar meson fields ~k1 and ~2 associated with particles having bare masses rn 1 and m2, respectively, and which interact through the interaction term =LPx = --?:¢1~p2:.

(2.9)

Thus ? plays the role of coupling constant. Being connected by the transformation (2.4), the systems described by the Lagrangian densities (2.1) and (2.5) must be equivalent. This means that the original system of two non-interacting real scalar fields ~b1 and ~b2 describing particles of masses #l and P2 must be completely equivalent to the two real scalar ~fields (k~ and ~k2 describing particles of bare masses ml and m2 and interacting drrough the interaction term (2.9). It follows t h a t / ~ and/t 2 are the dressed masses of the particles described by the fields ~1 and ~/2. Let us write g

= m j2 + A m j2,

j = 1, 2'.

(2.10)

Using eqs. (2.6) and (2.7) one finds that

am~ =

( p , z - p 2 ) sin 2 O,

Am~ = (/.t2 -/.t~) sin 20.

(2.11)

The purpose in what follows is to calculate Am~ and Arn~ using standard perturbation and renormalization methods and to compare the results with the exact values (2.11). 3. Mass Renormalization Conditions

From eq. (2.10) we have m j2 = i t2) - A m j ,2

j = 1, 2.

(3.1)

Substituting this into the Lagrangian density (2.5), one gets

=

2 21Z j=l

. 2.2. (0 //v."|~j,g@j,v'--~j

1

2.2.

+ ~1A m 22 ..i//2 2 :

(3.2)

The above Lagrangian density can be regarded as describing two real scalar fields @l and ~2 associated with particles of bare masses/'1 and #2, between which there exists an interaction described by the interaction Lagrangian density ~

= iA~2..I,2, a 2. 2 2. . . . 1 . e l . + ~ A m 2 .@2: - ~':@l~k2:.

(3.3)

48

M.

RAM

AND

N.

ROSEN

Since the bare masses of the particles have been taken here to be equal to the corresponding dressed masses, it is evident that the interaction ~ must not change these masses. The mathematical conditions expressing this fact, i.e., that there be no self-energy effects, are t

<

Xp)IRI%0)> = 0,

j = 1, 2,

(3.4)

where [~j(p)) represents a state of a free meson of mass/~j and momentum p. The matrix (1 + R) is that part of the S-matrix from which all vacuum processes have been eliminated. The state I~j(p)) can be described in terms of the normalized vacuum state I~o) and the creation operator a• (p) of a meson of mass #j and momentum p as follows: [ e j O ) ) -- aj~O)l~o>,

j --- 1, 2.

(3.5)

The energy Ep of the particle is given by the usual relativistic relation Ep = (p2 +/~)~,

(3.6)

and the expression for the S-matrix is

d4xl..,

S = 1+

d4x, S.(xl,.

--co

..

x.),

(3.7)

co

where

i" S, = ~ V { ~ e ; ( x l ) . . . ~ ; ( x , ) } .

(3.8)

Here x,, is the space-time 4-vector with components x ° = t, x 1 = x, x 2 = y, x a = z, T is the Wick chronological operator. One can therefore write that

R

d4xl n=l

J-co

d4x, R.(xl,.

. .

x,),

(3.9)

co

where

i, T'(~(xl)

~L~'{(xn)}.

(3.10)

The prime on the T in the expression (3.10) indicates that one must not consider the vacuum processes when expanding the chronological product according to Wick's theorem. Substituting the expression 0 . 9 ) into eq. (3.4) one gets for the mass renormalization conditions

(~i(P)[

d4xl.., nm

--CO

f See Schweber1), p. 512.

f; d'~x,Rn(xl . . . . x,)l~j(p) ) CO

= 0, j = 1, 2. (3.11)

THE MASS RENORMALIZATION METHOD

49

The eqs. (3.11) involve Am 2, Am2,/*2,/*2 and ?. Taking/*2,/,2 and ? as known, one can solve them for Am 2 and Am~ and the values obtained can be compared with the known values (2.11). It must be remembered that since the mass renormalization has been performed, one must use in all calculations the dressed masses/*j. In fact, the Fourier expansions for the positive and negative frequency field functions @(j+)(x) and @~-)(x) and for the chronological pairing $j(x)$j(y) are, after renormalization,

~,~+)(x)= &fko> oT)(x)

=

1

o (2cok) ~d3k

fk

d3k

e-ikxaj(k),

(3.12)

eikXa+(k),

(3.13)

@j(x) = ~5 +)(x) + Ip}-)(x),

I~]j(X)~j(Y) = - - - 4 (2~)

f

e - ik(x - y)

"2 2 d*k. k - #j + i~

(3.14) (3.15)

In the above, k is the energy momentum 4-vector and k ° = COk= (kZ+/*]) ~,

(3.16)

ej is a real positive infinitesimal parameter introduced by Feynman in order to make the integral (3.15) definite. This parameter will prove to be important. The creation and annihilation operators a + (k) and aj(k) satisfy the usual commutation rules [al(k), a+(k')] = 6tm6(k-k'), [a,(k), am(k')] = [a+(k), a+(k')] = O, 1, m = 1, 2.

(3.17)

4. Correspondence Rules In order to obtain Am~ and Am2 from eqs. (3.4), it is necessary first to calculate the matrix element <~(p)]R[¢j(p)>, where j = 1, 2. This can be done by using the method of Feynman diagrams and establishing correspondence rules between selfenergy diagrams * and the corresponding matrix elements. The derivation of these correspondence rules involves a simple calculation which, for the sake of brevity, will not be given here. The method used is the same as that of Bogoliubov and Shirkov 2). The correspondence rules needed are listed in table 2. They are based on table 1, which gives the general rules of correspondence between factors in the scattering matrix and elements of Feynman diagrams. It is important to note that because of the form of the interaction (3.3) which involves only bilinear terms in t It is to be n o t e d t h a t t h e only processes giving a n o n - v a n i s h i n g c o n t r i b u t i o n to <~j(p)[R[~t(p)> are the self-energy processes o f m e s o n s o f mass/zj, hereafter to be denoted by "/z~ m e s o n s " .

50'

M. RAM A N D N . ROSEN TABLE

1

Rules o f correspondence for factors in scattering m a t r i x a n d elements o f F e y n m a n d i a g r a m s FaVor of scattering matrix,

Element of Feynman diagram

~(X-)(X)

x°

O(l+)(X)

. . . . .

~(-)(X)

xO . . . . . . . . . . . . .

~2 +)(x)

.............

~'~(Xl)~/I(X2) .,1

--'*--

•

I

xl

~¢2(X1)~2(X2)

X1

--iy

(½JAm 2) (½JAm 2)

emerging external #1 m e s o n line •x

entering external /A m e s o n line e m e r g i n g external /z 2 m e s o n line

•x

entering e x t e r n a l / ~ m e s o n line

•

internal/~1 m e s o n line

•

internal /~, m e s o n line

x2

• ............

t - - J

Description o f element

......

X2

• ......

y vertex

0

Amx 2 vertex

63 . . . . . .

Am2 ~ vertex

TABLE 2 C o r r e s p o n d e n c e rules for m a t r i x elements o f self-energy processes in m o m e n t u m representation a) /~x m e s o n s e l f - e n e r g y p r o c e s s e s E l e m e n t o f the diagram

F a c t o r in t h e m a t r i x element

rc V T

•

. . . . . .

rc V T

•

/t 2 m e s o n self-energy processes

b)

)'

1

•

E l e m e n t o f the diagram

__

Factor in the m a t r i x element

---•

VT

•

VT

• ............

1

•

--

131

• ............

i

•

~2

•

(½idm~)

......

......

o ......

(½i~mD

- - - o - -

-

•

.

.

.

.

.

.

.

i7

i

•

®

-

i

- - - -

(½Mm])

O ......

•

.

(½i~m~) .

.

.

.

.

.

i7

a) I n addition to these correspondence rules o n e m u s t include in the m a t r i x element o f a n nth order process a factor 1/nt. b) W e denote by VT the space-time v o l u m e which h a s to be t a k e n to t e n d to infinity.

THE MASS RENORMALIZATION METHOD

51

the field operators, only two-meson lines may meet at any vertex, so that the selfenergy diagrams consist of elements all arranged in one line. As an illustration, a five vertex self-energy diagram of a / q meson is given in fig. 1. @,.

...... ®---® ......

-

Fig. 1. Fifth order/z1 meson self-energy process. As the 1/~1 and 1/8 2 terms appearing in table 2 play a crucial part in what follows, some discussion concerning their appearance is necessary. It is seen that these terms are due to internal #j (j = 1, 2) meson lines appearing in self-energy diagrams of pj mesons. As mentioned before, the self-energy diagrams consist of external and internal meson lines all arranged in one line. Because of energy-momentum conservation, the energy and the momentum of the virtual mesons must be equal to the energy and momentum of the external meson. In the evaluation of the selC-energy matrix dements this is equivalent to replacing the propagator 4-momentum k by the external 4-momentum p so that a term proportional to 1 Fy - p 2 _ p } + iej

(4.1)

appears for every internal P1 meson line (see eq. (3.15)). If the external meson is a #j meson, then p2 = pZo_p2 = pz. (4.2) Substituting this in the expression (4.1) gives the term proportional to 1/e~. As mentioned earlier, only the/~j meson self-energy processes contribute to the matrix element <~i(p)[Rl~by(p)>. One can therefore write

<4)j(p)lRl~i(p)> =

<~i(p)l ~l

f _~d4x, . . . f~ood4x, R,(xl , . . . x,)] ~j(p)>,

= (~i(p)k~= 1 f?ood4x,.., f~oodax,, ~ Rn(~,s)(xl .... x,Oicbj(p)>,

(4.3,

where R,(~)j is the operator corresponding to an nth order self-energy process of a /t~ meson. The subscript aj stands for an index or series of indices serving to identify the nth order self-enelgy process. From the form of the interaction term (3.3) and the expression (3.10) one easily sees that =

in

1

2 rj 1

2 rf

r 2s

(-)

~,._,(x._,)~, .... (x.)O(i+)(x.),

(4.4)

where J i , ./2 . . . . Jn-1 are indices that can have the values 1 or 2, j and j ' can also have the values 1 or 2 but j # j', rj is the number of Am 2 vertices in the self-energy

52

~. RAM ANON. ROSEN

diagram, rj, is the number of A m 2, vertices in the self-energy diagram, 2s is the number of y vertices in the self-energy diagram. Using the correspondence rules of table 2, one easily establishes that

fd4xl...fd'x.R.(.j)(xl

. . . . x.)l~j(p) >

in [1A~2~rjI'&A~2 ~n+rj--2(tj+ 1).,2(td--r£+1 ) n ! V ~ " ~ J I k~°'"J'!

Y

717

VT

Ep

~

(4.5)

#j --1~j.

\eft

Here tj is the number of internal gj meson lines. Use has been made of the following relations that can easily be proved: rj+rj,+2s

(4.6)

= n,

2 ( t j - r j + 1) = 2s.

(4.7)

From the expression (4.5) one sees that the self-energy processes of #j mesons having the same n, tj and rj have the same matrix element. We shall call the number of processes having the same n, ts and rj the multiplicity of the process and denote it by Nj = Nj(n, tj, rj). It can be shown that Nj:n,2"+2":-2(tJ+l)(tj+l](n-tj-2). \ ry / \ t j ~ r j

(4.8) /

5. Calculation of A m~ Using eqs. (3.11), (4.3) and (4.5) one can write ~VT~

1

N j i" (1A~a2~rJ[1h~a2~n+rj-2(t$+l)~2(tj-rd+ \ # j - I~j.]

\sd

=0.

(5.1) It is convenient to re-arrange the order of summation in the above expression. One divides all the #j meson self-energy processes into groups such that each group contains processes having the same tj, i.e., the same number of internal #j meson lines. One then performs the summation over all the self-energy processes by summing first over all processes in each group and then summing over all the groups. In this way one obtains

Ep

tj = 0

\ej oo

,n

tj "~ M l (.~A~2~rj[1Av~2~n+rj_2(tj+ l)~2(tj_rj+ l) / ~ " , j --q. ~2 . . . . j j ~ . . . . j'] rj=O n=2(tj+ l)--rj l'l! ( i )n--td--l(l__itJ 1 ×

-t- 2

= o.

(5.2)

THE MASS RENORMALIZATION METHOD

53

The first term in the summation, -7r

V T ~ i b + l (Zmj)2tj+l (1t

Ep

tj = 0

\Sjl

corresponds to the processes for which n = rj = tj+ 1. These processes have been separated, as they have to be summed over t3 only. Eq. (5.2) can be simplified to give

{(Am~)tj+t + ( - 1 ) b'+l ~ td=O

~

( - - i n) n~

,rj=O n=2(tj+ 1)--rj

(½Am~)r~(½Am~')n+rj-2(tj+l'

.

\#j - #j,]

) \~j/

= o. (5.3)

Since the above is true for arbitrary ej the coefficient of each power of be equal to zero, i.e.

(Am3)"+'+(-X)

tj

oo

Z

Z

(-1)"

1/ej must

(½ m3)"(½Am3,) " + ' - 2 ' ' + ' '

rj=O n=2(tj+ l ) - r j × ]72(b--rj+ 1) \ ~ 1

tj = 0, 1, 2 , . . . .

= 0,

(5.4)

These equations can be simplified by substituting in them the expression (4.8) for Nj. Some simple algebraic manipulation then gives

(-Arn]) 'j+l+

Z (--lf' r3 = 0

\

(Arn~f'

rj

2 ~ - Alqq,j2.,] 12j -- ltj, +

= O, t j = 0, 1, 2 . . . . .

(5.5)

This can be simplified further to give ]72 -

tj+l

Am} + 2 2 Am °'y,/ #j - #j, +

j = 1,2,

= 0,

(5.6)

so that, for all values of tj

A m~(#~ - #2) + (A m~)(A m~) = ]72, 2

2

2

2

2

A rnz (/~z -/~1) + (Am 1)(Arn2) =

(5.7)

]72.

AmZx and Am~. One obtains Am~ = ( # ~ - p 2z) sin z 0, Am~ -- (/~2_p2) sin z 0.

The eqs. (5.7) can be solved for

(5.8)

One sees that the expressions (5.8) are the same as (2.11), so that the mass renormalization process has led to correct values of Am~ and Am~. It is interesting to note that this result holds for all values of the coupling constant 7, which may therefore be as large as one wishes. If we had tried to solve the set of equations (5.4) or (5.5), starting with t i = 0, we would have found, of course, that te solution to the first one would satisfy all the others.

54

M. RAM AND N. ROSEN

6. Other Applications It has been found that the mass renormalization method leads to the correct values of A m 2 (j -- 1, 2) for the interaction considered and that this result is valid for all values of the coupling constant. We also saw that the calculation of A m 2 involved the I/e~ terms in a very special way and was easily carried out by dividing the pj meson self-energy processes into groups involving the same number of internal pj meson lines. This method may be immediately extended to other interactions and we shall consider it briefly for the case of quantum electrodynamics. (a)

. ~ Xl

r x2

(b) Fig. 2. Electron self-energy processes up to the second order in the coupling constant. In fig. 2 are shown the fermion self-energy processes up to second order in the coupling constant. The smooth lines represent electron lines, while the wavy ones are photon lines. The mass renormalization condition * up to second order in the coupling constant is ( p ) + a ( A m ) = 0.

(6.1)

Here ~ ( p ) represents the contribution of the part between x 1 and x 2 in fig. 2(a) (p is the m o m e n t u m of the external fermion), a is a constant and A m is the change in the mass of the electron due to the electromagnetic interaction. The numerical coefficients have been determined so as to take into consideration the 1/n! factor appearing in the series expansion of the S-matrix, and the multiplicity of the processes. The electron self-energy p r o c e s s e s u p to fourth order in the coupling constant are given in fig. 3. The processes have been divided into two groups; group (a) includes all proper second and fourth order processes, i.e., those not involving "free" internal electron lines (by definition, a "free', internal electron line is an internal electron line such that, when cut in two, it divides the process into two disconnected processes); group (b) includes all fourth order processes involving one "free" internal electron line. Here too, because of conservation of energy and m o m e n t u m it can be shown that the contribution of a "free" internal electron line to the matrix element of the electron self-energy process is proportional to 1/e, where e is a real positive infinitesimal arbitrary number. We shall write this contribution as B(p)/e (B(p) is a function of the external electron m o m e n t u m p). The mass renormalization condition for the The photon mass renormalization condition can be set up in analogy with that of the fermion.

THE MASS RENORMALIZATION METHOD

55

electron up to the fourth order in the coupling constant is ~*(p) + {l" ~ (p)-]2 + 2~(A m) £ (p) + [-~t(dm)-]2} B(p) = 0.

(6.2)

Here ~*(p) is the contribution of the proper self-energy diagrams, i.e. the diagrams in group (a). The numerical coefficients of the terms in curly brackets take into account the 1/n! factor which appears in the series expansion of the S-matrix, and the multiplicity of the processes.

(a)

(b)

Fig. 3. Electron self-energy processes up to fourth order in the coupling constant.

Since e is arbitrary, eq. (6.2) leads to ~*(p) = 0,

(6.3)

~ (p)]2 +2~(Am) E (P)+ [ ~ ( d m ) ] 2 = 0.

(6.4)

The last equation can be factored and is equivalent to eq. (6.1). We see therefore that the condition obtained from the term involving 1/e is automatically satisfied if we take account of the second order mass renormalization condition (6.1). This result is similar to the one we obtained in the meson case where the infinite set of eqs. (5.5) for tj = 1, 2 . . . were automatically satisfied if eq. (5.5) for tj = 0 was satisfied. The main difference of course is that we have shown this to be true in quantum electrodynamics only up to fourth order in the coupling constant. If we consider all the electron self-energy processes to all orders, arrange them in groups such that every group contains processes with the same number of "free" internal electron lines, and then sum over all the processes, first summing over

56

M. R A M A N D

N. ROSEN

processes in individual groups and then summing over all groups, the mass renormalization condition one gets will consist of a series in powers of 1/8 which is equal to zero. Since e is arbitrary, the coefficients of the different powers of 1/5 must be equal to zero separately, f f we denote the coefficient of (1/5) ° by ~ * ( p ) then Z*(P) = 0.

(6.5)

~ * ( p ) is in fact the contribution of all the proper electron self-energy diagrams. The condition (6.5) is exactly the mass renormalization condition usually derived in quantum electrodynamics by other methods a). As mentioned above, we have shown that the condition (6.5) insures that the coefficients of the other powers of 1/5 are zero, only to the fourth order in the coupling constant. Evidently the method just applied to quantum electrodynamics may be used similarly in other cases.

References 1) S. S. Schweber, Introduction to relativistic quantum field theory (Row, Peterson, Evanston, Illinois, 1961) 2) N. N. Bogoliubov and D. V. Shirkov, Introduction to the theory ofquantized fields (Interscience, New York, 1959) 13. 239-257 3) J. M. Jaueh and F. Rohrlich, The theory of photons and electrons (Addison-Wesley, Cambridge, Massachusetts, 1955) 13. 219