P H Y S I C A L R E V I E W D V O L I J M E 2 0 , N U M B E R 4 1 5 A U G U S T 1 9 7 9

Explicit solutions for the 't Hooft transformation

N. N. Khuri and Oliver A. McBryan* Rockefeller Uniuersity, New Yoric, New York 10021

(Received 15 March 1979)

The 't Hooft transformation is a change of variables of the renormalized coupling constant to a new coupling parameter such that the resulting Gell-Mann-Low (GML) function is a low-order polynom~al with coefficients independent of renormalization scheme. Explicit expressions for thls transformation are given for (4~~),, d = 2,3,4, field theories, QED, and quantum chromodynamics. The solutions are expressed in terms of the original GML function P(g) and their properties are given in some detail. In several cases these transformations are shown to be valld up to the first nontrivial zero of P(g).

I. INTRODUCTION

In a renormalizable o r superrenormalizable field theory one can use different mass-indepen- dept renormalization schemes. Each scheme will give a different definition of the renormalized coupling constant g and the corresponding Gell- Mann-Low (GML) function p ( g ) that appears in the renormalization-group equations. Thus if g is the coupling in one scheme with a corresponding P(g) , another scheme will yield g' with a GML function P f ( q 3 , such that g ' - ~ ( g ) =g + o ( ~ ' ) . The functions p ' and p a r e in general different func- tions, but the f i r s t two coefficients in their per - turbation expansion mus t be identical. F o r exam- ple in (G4), theory, if P1(g) = a l g + a z g 2 + O(gY) and P f ( g ' ) = a;g1+ + o ( ~ ' ~ ) , then a l = a; and n2 = a;.

't ~ o o f t ' has suggested that one can exploit this f reedom in the choice of g even fur ther and choose a new coupling parameter g, such that the c o r r e s - ponding P,(g,) z a l g R + azgR2 (for the above case) and thus has only two t e r m s in i t s expansion in g,. The t ransformation g-g,, which we shal l cal l the 't Hooft transformation, is defined to sa t - isfy P(g)a/ag = pR(gR)a/agR s o that i t p r e s e r v e s the fo rm of the Callan-Symanzik equation when written in t e r m s of g,.

In perturbation theory the solution to the 't Hooft t ransformation exists. ' Namely, given P(g) =Gang", and writing g, G ( g ) = g + C r n g " , one can determine a l l the r's in t e r m s of the a ' s . The following question immediately a r i s e s : Given a P(g) does a nonsingular invertible g, = G (g) exis t with P,(g,) = a i g R + azqR2 for some interval 0 G g Ggl? If so, how la rge is the interval 0 < g <gl and what a r e the general propert ies of G(g) and aG/ag i n this domain?

In this paper we show that the 't Hooft function G ( g ) can be given explicitly in t e r m s of P(g) and we shal l r igorously study its propert ies fo r sev- e r a l field theories ($4)2, (G4),, (G4),, QED, and quantum chromodynamics (QCD) .

In addition to the original application of this t ransformation by 't ~ o o f t ' there a r e o ther appli- c a t i o n ~ , ~ but in this paper we sha l l concentrate only on problems related to the existence of G ( g ) , i t s propert ies , and the location of i t s leading s in- gular i t ies . One of our main objectives i s to c l a r - ify the relationship of the f i r s t nontrivial ze ro of p(g), when such a z e r o exis ts , to the z e r o of PR(gR) which we ca l l the 't Hooft ze ro . We a l so show what happens in c a s e s such a s QED and QCD where p,(a,) has no positive ze ro .

We s t a r t by t reat ing the c a s e of $"field theory in 2 and 3 dimensions. Here P(g) has the fo rm, f o r s m a l l g ,

P(g) = a l g + nlg2 + ~ ( g " . (2.1)

It i s important to recal l that in these two c a s e s the s igns of a , and a, a r e such that

In fact, one usually resca les g such that a , = -1 and a z = +1 a s in Ref. 3.

The 't Hooft t ransformation f o r this c a s e is de- fined a s follows:

gR z ~ ( g ) =g + 0 ( g 2 ) .

One then defines P,(g,) a s

and looks for a G ( g ) such that

PR(BR) = ~ I B R + a2gR2 . (2.5)

Clearly in the domain where G ex is t s and where a ~ / a g > O we have

The problem is now, given p(g) in some interval

20 - 88 1 @ 1979 The American Physical Society

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0 <g<g,, can one find a G ( g ) that sat isf ies Eqs. (2.3), (2.4), and (2.5)? More simply, what we have to do is study the solutions of the f i r s t -o rder non- l inear differential equation,

such that G (x) = x + 0(x2) a s X- 0 . This equation can be solved exactly in t e r m s of P(g). To s e e this we f i r s t wri te f(x) =G(x)/x and (2.7) becomes

with

b = az/ai . (2 .lo)

Note that k(x) >O f o r 0 < x < g * where g * is the f i r s t IR zero of P(g) , p(g*) = 0. Writing

the differential equation (2.8) reduces to a l inear equation

Using s tandard methods and converting back to our original function G ( g ) we get, using Eq. (2.3),

We reca l l that in this c a s e b = (a2/al) <0, s o the denominator does not vanish f o r 0 <g<g*. It is now a m a t t e r of s imple a lgebra to calculate d ~ / d g explicitly also and one gets

One should note that f r o m Eq. (2.1) i t follows that all the integrals in Eqs . (2.13) and (2.14) above a r e convergent a t x = 0.

Mathematically, there a r e now two distinct c a s e s t o study: Case ( I) , where P(g) s t a r t s out negative n e a r g= 0, remains finite fo r some interval 0 <g <g* and develops a z e r o a t g = g * , P(g*) = 0; p ( g ) <O f o r 0 <g<g*. We shal l assume that the z e r o is such that fdx /o(x) diverges a s g -g* . This is the actual physical case f o r ( $ I ~ ) , and (c)~),. Case (2) will cover the situation where P(g) has no finite ze ro except a t g = 0 , and we shal l distinguish be-

tween the two possibilities g dx/p(x) finite o r di- vergent .

Given the explicit f o r m s of G(g) and G t ( g ) i t is easy to derive the propert ies of this function f o r both cases .

Case (1). (a) G(g) =g + 0(g3) with no g2 t e r m f o r s m a l l g. (b) G(g) ex i s t s and is bounded f o r 0 < g < g * and

a s g -g*,

where g: is the ' t Hooft ze ro defined by the non- t r ivial root of p,(glT) = a l g x + a Z g i 2 = 0 .

(c) Gt (g) >O f o r 0 s g < g * , and G maps the inter- val 0 < g < g * in a one-to-one manner onto 0 Gg ,

< g i = -al/a2. (d) If P ( g ) has a f i r s t z e r o g * neares t the or igin,

then G(g*) =g$, the ' t Hooft ze ro . However, the existence of the ' t Hooft z e r o does not imply the existence of a z e r o in P(g) a s we shal l s e e in dis- cussing c a s e (2).

(e) Finally, i t i s important to study the behavior of G a s g -g*. To do this we assume that p (g) has a s imple z e r o a t g * and wri te

P ( g ) p w ( g - g * ) , g - g * (2.16)

where w >O i s the slope of P a t the fixed point. It is sirnple to check f rom Eq. (2.14) that a s g-g*,

Therefore, in general , G '(g) would vanish o r di- verge a s g-g* unless by some accident w - -a,. This accident could happen for (G4), where the value w = 1 = -a1 i s consistent with the bes t cal- culations of the slope.4 However, in ( c # J ~ ) ~ the bes t value f o r w r 0.782, while -a, = 1, which gives a G1(g) that vanishes a s g -g* , even though i t van- i shes slowly, i.e., - (g* -g)OSz7. Thus, in general, G ( g ) will develop a branch point a t g = g * .

Before we d i scuss case (2), we would like to re - m a r k on the universality of the slope of @ ( g ) a t the fixed point. It has been shown5 that f o r two different renormalization schemes ap/ag/,,,* = 8pf/agflg,=,*.. However, the proof of th i s fact depends on the assumption ag'/ag #O a s g -g*. This assumption is clear ly not always t rue f o r ag,/ag as g-g*. In fact, we can wri te

and thus we can rewri te Eq. (2.17) a s G'(g) = (g* - g ) ( W ~ / W ) - i . Now one can s e e that Gt (g) #O a s g -g* can only happen if w, = w and the assump- tion about G f ( g ) begs the question. However, i t might s t i l l be interest ing to seek a physical ex-

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planation f o r the fact that the quantity (wR/w) - 1 = 0 f o r ( Q 4 ) 2 and (wR/w) - 1 a 0.27 f o r ( Q 4 ) , , thus smal l in both c a s e s . As an approximation w z w, is only 20% off f o r ( @ 4 ) 3 and is almost exact f o r ( @ 4 ) 2 . Also, in ( @ 4 ) 4 the approximation w r w, s e e m s to be good f o r a ce r ta in Bore1 sum of ~ ( g ) . ~

Case (2). In this c a s e ~3 remains negative and has no finite z e r o s f o r O<g<m. Then i f Lmdx/P(x) is convergent we get a G ( g ) which is bounded and monotonically increasing with g . It maps 0 <g < .o

onto some interval 0 <gR <gEax such that g:= <g$ = - l / b . The function G ( g ) inc reases but never reaches the value of the ' t Hooft z e r o gi. The left-hand s ide of Eq. (2.4) never vanishes a s g var ies in 0 <g <m.

F o r the c a s e where al p d x / p ( x ) - +m, we again get a monotonically increasing function G ( g ) f o r O<g<m where now a s g-m, G ( g ) -gR= - l / b . The interval 0 s g < m i s mapped in a one-to-one manner onto 0 S g , < g i = - l / b .

We know that ( @ 4 ) 2 , ( @ 4 ) 3 both have a nontrivial IR z e r o g* of P(g) . We only discuss c a s e (2 ) to make c l e a r what happens to the solution of Eq. (2.13) when P has no z e r o and to s t r e s s the fact that the exis tence of the ' t Hooft z e r o does not necessar i ly imply a ze ro in P(g) . However, if P(g) i s known to have a f i r s t ze ro a t g=g* that z e r o must lead to the 't Hooft z e r o under the t ransformation gft = G ( g * ) .

Finally, fo r completeness, we mention what hap- pens when @ ( g ) develops a finite ze ro such that

d x / ~ ( x ) converges a s g -g* . In this c a s e G ( g ) will be bounded and monotonically increasing f o r 0 - 'g Sg* with G(g*) < - l / b , and the value of the ' t Hooft ze ro is not reached. However, now i t is evident f r o m Eq. (2.14) that G ' ( g ) " [b(g)]-' a s g -g* and will diverge in such a way that the right- hand s ide of Eq. (2.4) remains finite a s g-g*. Although this is an unphysical c a s e f o r ( Q 4 ) 2 and ( Q 4 ) , a branch-point-type behavior in P cannot be ruled out in general in other field theories and can be made consistent with the renormalization group.?

111. CASE OF (q54 )4

In four dimensions the GML function has a po- wer -se r ies expansion of the f o r m

where

Proceeding a s before we define

with

The coefficient of g 2 in (3.3) denoted by r, will turn out to be a f r e e parameter and appear a s an integration constant below.

As before, we seek solutions of the differential equation

Unfortunately, unlike the previous case , we cannot wri te out the solution explicitly f o r this case . However, we can do almost a s well by getting an implicit fo rm for the solution.

Omitting the algebra we get

where q i s the constant that appears in Eq. (3.3). It is easy to check that G defined by (3.7) is the solution of Eq. (3.6) with initial conditions speci- f ied by Eq. (3.3).

Although Eq. (3.7) is not explicit, i t is more than adequate f o r determining the relevant p roper t i es of G ( g ) star t ing f r o m g = 0 . To do this we mus t keep in mind that in this c a s e b = a,/a2 <0.

Again we have to consider two c a s e s separately: case ( I ) , where ,i?(g) s t a r t s positive f o r g>O and develops an ultraviolet (UV) ze ro a t g=g,, p(g,) = 0, -P f (gm) <m; c a s e ( 2 ) , where P(g) has no z e r o s fo r finite g = 0 and i s positive f o r 0 <g <a.

Case (1) . G ( g ) = g f o r smal l g >O and increases monotonically, G ' ( g ) >O in the interval O <g <g,. As g -g , f r o m below the right-hand s ide of Eq. (3.7) diverges, this can occur only if the argument of the logarithm on the left vanishes and we get a s g -g , f r o m below

Thus again G ( g ) maps the interval 0 --'g s g , in a one-to-one manner onto 0 g, ~ g , " , where g; is the z e r o of the 't Hooft O R , P,(g;) = a2g,"' +a3g23 = 0 .

However, again a s g -g,, G ( g ) will in genera l develop a branch point. The nature of this branch point will depend on the slope of P at g=g,. As before, we take

and Then f r o m Eq. (3.7) we get a s g -g , f r o m below

884 N. N . K H U R I A N D O L I V E R A . M c B R Y A N 20 -

G ( ~ ) + l / b % (g, -g)az'Wb, g = g , . (3. l o ) However, this i s not the function that should be

The quantity a2/wb = aZ2/wa3 >O for (G4),, w <0, a3 <O. It i s instructive to express Eq. (3.10) in t e rms of the slope of the 't Hooft function PR(gR). We define

This gives us

and

G ~ ( ~ ) = ? ( ~ . - ~ ) ~ R / ~ - ' , g-g.. (3.13)

As in the (44)2 and ($4)3 cases, G1(g) will in gen- e r a l vanish o r tend to infinity a s g-g,. Only if w, = w will G be nonzero a s g -g,. The ' t Hooft transformation by itself obviously gives no relation between w and w,. However, if from some additional physical input one can show that G ' ( g ) vanishes slowly as g -g,, then (wR/w) -1 << 1 and one gets w, = w. For the specific Borel summation method used in Ref. 2 the function P,(g) and i t s corresponding G,(g) seem to satisfy this property and this explains the approximate agreement between w, and w, in that case to with- in 20%.

Case (2). Here P(g) has no finite zeros. G ( g ) increases monotonically in 0 c g < . o . The behavior as g-.o depends on whether dx/p(x) converges o r diverges. In the f i r s t case lim,,G(g) =ggax and gzax < (-l/b), thus 0 G g s .o i s mapped onto 0 sgR sgEDw In the second case lim,,G(g) = - l /b =gz , and 0 sg< .o i s mapped onto 0 6 g R < g i .

Finally, we should note that in case (1) we only discussed the consequences of a simple zero in @(g) at g=g,. In general, one could study other possibilities including the situation where the zero a t g=g, i s a branch point and l d x / ~ ( x ) i s convergent a s g-g,. For the sake of brevity we limited ourselves to cases (1) and (2).

IV . QED

This case differs from the 44 cases considered before because the relative sign in the f i r s t two t e rms of the Gell-Mann-Low function i s positive not negative. This makes a significant difference in the behavior of the 't Hooft transformation func- tions.

The renormalization-group function P(a) i s given up to third order in a by De Rafael and ~ o s n e r ,

transformed by the ' t Hooft transformation, since in the Callan-Symanzik equation for QED' the op- e ra tor ap (a )a / aa appears. One should thus look for a ' t Hooft transformation f o r p s a P ( a ) which transforms to a two-term expansion. We shall do this below but f i r s t i t i s instructive to consi- der the transformation of p a s given by (4.1).

We seek a transformatioil a - a, = G(a) such that

The equation we have to solve i s

with G(a ) = a + 0(a2). This i s the same a s Eq. (2.7) in the ($*)), case with g replaced by a. The solution will be the same a s in Eq. (2.13), but now b = a2/ai >0, and the denonlinator can vanish. The interesting case to study i s again that where a f i r s t zero develops a t a = ao, P(ao) - 0, such that

dx/p(x) diverges as a - ao. Then i t follows from Eq. (2.13) that C ( a ) will diverge at some a = a, with a, < a o , and given by

It i s fairly easy to check that for the above case a solution of Eq. (4.4) always exists with a1 < a O . As a - a , , G(a ) will develop a simple pole, G ( a ) - ( a1 -a ) " , ,=a1. T h e d o m a i n O < a < a l i s m a p - ped in an invertible manner onto 0 < a R <.o but the zero of P(a) i s outside this domain. h here i s nothing, however, to prevent one from studying the solution in the region a > a , . The second branch of the function G(a ) will map the domain a , < a < a. onto -a < a , < -a2/al. PR(aR) in this case has no positive zeros, but i t has a negative zero at a! = -a2/al <O.]

The behavior in the case where dx/p(x) con- verges a s a! - ao, P(ao) = 0, i s also worth noting here. In this case Eq. (4.4) may not have a solu- tion for 0 < a l < a o . The resulting G(a) will vary in some interval 0 a, 6 agaX for 0 a s ao. AS a - a o , G '(a) = @(a)]- ' and will diverge at the zero. Nevertheless, the whole interval 0 s a 6 a, i s mapped in a nonsingular manner.

The physically relevant 't Hooft transformation to consider, however, i s the one for the function ap (a ) =p (a ) since this i s the combination that ap- pears in the Callan-Symanzik equation. We want a transformation such that

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where

with a2 = 2/38, a3 = 1/28', and a3/a2 z b >O. The transformation we want a , =G(a) = a + 0 ( a 7 ) i s now defined by the differential equation

This is the same as Eq. (3.6) for (44)4 , except here b sa3/a2 >O. The solution i s a s before

Again the interesting case to study i s the case where ?(a) develops a zero at some a = a,,, a , being the zero nearest the origin, and S,O dx/p(x) diverges as a -a , .

The solution c ( x ) s tar t s at zero and increases monotonically and c ( x ) will diverge as x - a , < a o , where a , will be given by

If p(a,,) = 0, and k o d x / @ ( x ) diverges, this equation always has a solution for some a , < a o . To see this, one has to notice that the right-hand side of (4.8) increases monotonically from - a - + a as a varies in the interval 0 G a c a,,. It i s easy to check that a s a - a , , G ( a ) = ( a l - and develops a branch point a t a = a ,.

Hence, because of the relative sign of the f i r s t two coefficients, the result in QED i s not a s useful a s in the (44)4 case and we do not get a nonsingular mapping in a domain large enough to reach the f i r s t zero of p(a) , if such a zero exists.

The case where p(a) has no finite zeros can similarly be studied and the result will depend on whether Jmdx/P(x) is finite o r diverges.

Actually one can do better in QED if one gener- alizes the ' t Hooft idea and transforms to a new variable a , that gives a pR with three te rms, namely,

formation to consider. The main thing to notice i s that in the scheme

of Ref. 8, the f i r s t three coefficients as given in Eq. (4.1) give a polynomial which has two nonvan- ishing real roots, one positive and one negative.

The problem i s to study the differential equation

The solution with initial conditions H(x) = x + r2xz + 0(x3) will be given by

d x S - L + L 2 L -L-% =[ (@(x) x' arX ) a2 l na + % , (4.12)

where (-A,)" and (-A,)-' are the two real roots of the polynomial 1 + (a3/u2)x+ (a4/a,)x< and with a4 a s given by Ref. 8, X l >O and A, <O,

One can now check the properties of H(x) from Eq. (4.12). We do this f i r s t for the interesting case where p(a,,) = 0 for some a = a,,, a,, being the zero nearest the origin. Then again H(a) s t a r t s at zero for a = 0 and increases monotonically until 1/H becomes equal to - X , , where ( X , ) - l i s the nega- tive root of the polynomial in (4.10). We a re as- suming of course that k dx/p(x) diverges as a -a,, from below we get

1 l i m ~ ( a ) = - - - = a : , X,<O a-a,,

(4.14) A,

where (-A,)- ' can be calculated from Eq. (4.13) and i s a zero of pR(aR) .

In QED there i s a respectable conjecture that if p(ao) = O this zero i s an essential zero.9 If such i s the case then aH/aa as a - a o will also develop an essential zero. Otherwise, if p has a simple zero at a =a,, the properties of aH/aa a s a - a , will be the same as in the previous section.

What we have gained by adding an additional te rm in Eq. (4.11) i s that now the full domain up to the f i r s t zero 0 =S a < a,, i s mapped in a nonsingular manner onto 0 a , < a:, where a! i s now the positive zero of the three-term $,(aR) given in Eq. (4.10). The properties of H(a) in the case where a has no finite zeros can be deduced a s be- - 4 @,(aR) z a2aR2 + a3aR3 + adaR . (4 .lo) fore.

The coefficient a4 willin this case depend on the renormalization scheme of the original p(a) . This

V. QUANTUM CHROMODYNAMICS

i s a distinct change in the original motivation of This case i s similar to QED. Asymptotic free- ' t Hooft which was to express the GML function in dom here does not make a difference, the relevant a way that would be independent of a renormaliza- property i s the relative sign of the f i r s t two co- tion scheme. However, it i s still a useful trans- efficients of @ ( a ) . Writing g2 = a one has1

886 N . N . K H U R I A N D O L I V E R A . M c B R Y A N

where f o r the s tandard c a s e

a 2 <O, a 3 > 0 , and b =a3/a2 >O . (5.2)

In the notation of Ref. 1

where Nf is the number of f lavors , and we consi- d e r the case

s o that we not only have asymptotic freedom but

F o r Nf such that >Nf >% we s t i l l have asympto- t ic f reedom but b = a3/az <O. We shal l briefly only d i scuss the s tandard case , Nf c 8.

This is already covered by the c a s e of QED given in Eq. (4.7). Note that the rat io a2//3(x) is always positive in the neighborhood of the origin regard less of asymptotic freedom.

One obtains aR=G(cr ) given essent ial ly by Eq. (4.8) and PR(aR) = a 2 a R 2 + a3aR3. In QCD i t is a t l eas t hoped that P ( a ) has no finite z e r o s . We dis- c u s s this possibility f i r s t . Then, a s before, G ( a ) will inc rease monotonically a s c! inc reases f r o m zero . It could diverge at some point, a = a , , given

where r2 is an a r b i t r a r y integration constant. Thus, if Jmdx//3(x) is divergent, one can always find a solution of (5.5) f o r any finite r 2 , fo r some positive a , . The region i n which the 't Hooft t ransformation is nonsingular is 0 6 a, < a , , and a s before a s c! - a , , G(x) = ( a , - a)-'''.

On the other hand, if t d x / p ( x ) , c >O, converges then one can always choose I r , l l a rge enough s o that Eq. (5.5) has no solution. Then one obtains a 't Hooft mapping which is nonsingular fo r the whole interval 0 6 a <m.

The case in QCD where P(a ) develops a z e r o f o r some a = a. can of course be easi ly handled and the resu l t s a r e almost identical with the discus- sion in the previous section following Eq. (4.8).

Finally, f o r # < N ~ <? we have b <0, and a situa- tion analogous to the ( @ 4 ) , case.

ACKNOWLEDGMENTS

T h i s work was supported in par t by the U. S. Department of Energy under Contract No. EY-76- C-02-2232B.*000. The work of Oliver A. McBryan was supported in par t by NSF Grant No. DMR-77- 04105.

"Permanent address: Department of Mathematics, Cornell University, Ithaca, N. Y. 14853.

l ~ . ' t Hooft, Erice Lectures, 1977 (unpublished). can use some of the identities obtained via the

' t Hooft transformation to check the self-consistency of approximate Borel summation methods such as the one recently given by the first-named author. See N. N. Khuri, Phys. Lett. g, 83 (19791.

3 ~ . A. Baker, J r . , B. G. Nickel, and D. 1. Meiron, Phys. Rev. B u, 1365 (1978).

*J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977).

5 ~ a v i d J. Gross, in Methods in Field Theory, edited by R. Balian and J. Zinn-Justin (North-IIolland, Amster- dam, 1976), pp. 177-179.

6 ~ . N. Khuri, Rockefeller University Report No. COO- 2232B-169, 1979 (unpublished). The assumption G ' ( g ) * 0 a t g = g, was made in that paper. Without that assumption the identification of w with uR cannot be made. However, for the specific Borel sums con- sidered % * w,, to about 15%, so G ' (g) vanishes slowly a s g-g,. One of us (N.N.K.) would like to thank C. Itzykson and M. Creutz for helpful comments on this point.

7 ~ e e for example R. Oehme and W. Zimmermann, Phys. Lett. 3, 314 (1978).

%. De Rafael and J. L. Rosner, Ann. Phys. (N.Y.) 82, 369 (1974).

$s. L. Adler, Phys. Rev. D 5 , 3021 (1972).