PHYSICAL REVIEW D VOLUME 37, NUMBER 12 15 JUNE 1988

Calculating the two-loop effective potential of the Weinberg-Salam model

Hamad AlhendiDepartment of Physics, College of Science, King Saud Uniuersity, P.O Box.2455, Riyadh 11451,Saudi Arabia

(Received 30 November 1987; revised manuscript received 21 March 1988)

A simplified method, for calculating higher-order corrections to the effective potential, based on

the loop expansion of the renormalization-group equation of the full effective potential, is used tocalculate the two-loop effective potential of the Weinberg-Salam model. Some remarks on the re-

sults and their further extension are discussed.

I. INTRODUCTION

The formalism of the effective potential, ' which gen-erates the connected one-particle irreducible Feynmangraphs with vanishing external momenta, has proven tobe a convenient tool in the investigation of the vacuumstructure of the spontaneous breakdown of the gaugesymmetry of renormalizable field theories. However,in application, the full effective potential ( V,tr) is general-

ly not known exactly, and one, therefore, may resort tosome approximation methods to obtain a truncated ex-pression. Unfortunately, while it is sometimes importantto include higher-order corrections, the application ofconventional techniques to the calculation of V,ff

beyond the one-loop level, except for trivial cases, is adifficult task and computationally tedious. The need thusarises to have a simple computational technique.

An important property of V,z is that it satisfies therenormalization-group equation (RGE). This means, ifthe renormalization-group (RG) functions are known,then, in principle, V,z could be obtained from the generalsolution of the RGE (Refs. 3 and 8). One, however, canstill employ loop-expansion techniques to the RGE evenif the RG functions are known only to some truncated or-der."

Recently the RG functions for a general renormaliz-able gauge theory with semisimple group have been cal-culated at the two-loop level, ' '" and even beyond. ' Inthis paper we shall make use of these calculations to ob-tain the one- and two-loop V,z of the Weinberg-Salam(WS} model, '

by applying the loop expansion to theRGE of V,z. This method, as will be seen, is less in-

volved compared to previous calculations using conven-tional methods, ' and ensures that a correct expressionfor V,z is obtained for the corresponding RG functions ofthe model. For our purpose it is sufficient, in this paper,to assume the existence of the expansion of V,z in powersof A'.

To some extent, the present work is an extension of theapplication of the renormalization group to the effectivepotential originally initiated in Ref. 3, and applied in Ref.9 to the calculation of the one-loop effective potential ofWitten's SU(5} model. " To our knowledge such calcula-tions have not been fully carried out in the case of the WSmodel by use of the RGE and the loop expansion, andprevious calculations' using conventional methods be-

sides being tedious are, in fact, incomplete, and couldlead to different parametrizations. ' Although, in this pa-per, we will be mainly concerned with the calculation ofV,~ of the WS model, the method is not limited to it.Throughout this paper we use the Weinberg —'t Hooftmass-independent renormalization scheme' and employthe Landau gauge.

In Sec. II the RGE of the full effective potential in themass-independent subtraction scheme is introduced, andthen the loop expansion is applied. A recursion relationfor the nth contribution to V,z in terms of the classicalpotential is obtained. In Secs. III and IV the one- andtwo-loop effective potential of the WS model are calculat-ed. Finally Sec. V consists of some remarks on the re-sults, and further extensions and possible uses.

II. THE RENORMALIZATION-GROUP EQUATION

For simplicity, we consider in this section a theory of asingle scalar field interacting in a renormalizable waywith a gauge field. The effective potential V,tt($, ) is afunction of the classical field P, given by3

(2.1)

where I'"'(0) is the one-particle irreducible (1PI) n-pointGreen's function calculated at zero external momenta. Inthe mass-independent scheme of Weinberg and 't Hooftthe renormalization-group equation (RGE) for 1'"'(0)takes the form

M +p P & 2 +P& +Pg +ny I "(0)=0(n)

(2.2)

and the corresponding RGE for the full V,s(P, } is

,M+1"P, +P»+P, ,=0, (23)

where M is a renormalization mass parameter, and p t,P

Pz, P, and y are renormalization-group functions formass, scalar, and gauge couplings, respectively, and y is

37 3749 1988 The American Physical Society

3750 HAMAD ALHENDI 37

the anomalous dimension, which may be perturbativelycalculated. It will be convenient to introduce a morecompact notation for the parameters of the theory.Define

k =((M, i, ,g),

V(0) 2@0~+ ( @i'@)22

The vacuum expectation value of 4 is

U =&@'/X .

Introducing a four-component real field (t) by

(3.1)

(3.2}

so Eq. (2.3) can be abbreviated as

a a aMaM+'P a, +'~ay, V,(t((}(,) =0, (2 4} where

(3.3)

where 5~ equals p for mass coupling and 1 otherwise.In the loop expansion V,z is given by

4$2 g p2 (3.4)

V V(0)+ ~ g(n) V(n)e8'

n=1

Eq. (3.1) becomes(2.5)

(3.5)

where V' ' is the tree-level potential. Similarly the RGfunctions have the expansion

The masses of the physical Higgs boson and the vectorgauge boson are

p y g(n)p(n)

n=1

y g(n)~(n)

(2.6)

(2.7)

mH —2A,U, M~ ——Tg 2U2 2 2 ] 2 2

M2 ( (g2+g2 )U2(3.6)

n=1

where X'n' is the n-loop contribution to X. By substitut-ing Eqs. (2.5), (2.6), and (2.7) in Eq. (2.4} and comparingpowers of h, one gets mf

gf=U

(3.7)

where g) and g2 correspond to the U(1) and SU(2) gaugecouplings, respectively. The Yukawa couplings are

M V"'+D V' '=0 for A order,aM

M V' '+D V '+D V' =0 for A orderaM 2 1

(2.8)

(2.9)

where f is the corresponding fiavor.We define, as renormalization scale t,

t= —,'ln( —q /M)4 ) (3.8)

and

M V'n'+D V' '+D V"'+ +D V'" "=0BM n n —1 1

for fi" order, (2.10}

where the differential operator D is defined by

D =5 p™a +y' 'p, , m =123,. . . .

(2.1 1)

and write p),——d k, /dt.

p

The one-loop p functions of the model are'

8m PI"=6k, —( —,'g, + —,'g2 —6g, )A,

+ 8g1+ 4g1g2+ 8g2 —6g, ,

16m Pg" (4, N —", )g2,—— —

16% P~g =g(( 2gt )2g ) 4g2 )2n(1) 9 2 ]7 2 9 2

(3.9)

(3.10)

(3.1 1)

(3.12)Equation (2.10}gives a recursion formula for the calcula-tion of the n-loop contribution to V,~ in trems of the RGfunctions and the tree-level potential. In particular, thecase n =1 is the usual result for the one-loop V,(t (Refs. 3and 9). On the other hand, Eq. (2.9) is the desired one forthe two-loop correction.

In the following sections we will apply these recursionrelations to the calculation of the one- and two-loopeffective potential of the WS model. Where no confusionmay arise we shall omit the subscript c from the classicalfield P, .

III. WEINBERG-SALAM MODEL:ONE-LOOP POTENTIAL

To fix our notation, we use the usual Lagrangian withone Higgs doublet. The Higgs potential is

P()) 2~() )

P

where

(3.13)

, (g (+3g 2 4g,')—64m

(3.14)

is the one-loop anomalous dimension in the Landaugauge which can be directly read from the coefficient of A,

in the scalar p function. We use N=3 for the number offamilies if not specified. We have neglected all Yuakawacouplings of light particles and the corresponding func-tions as well as radiative corrections due to the scalarmass term. However, the inclusion of radiative correc-tions will be considered at the end of this section.

Now, applying Eq. (2.8) to Eq. (3.5) using Eqs.(3.9)—(3.14) one gets, for V"',

37 CALCULATING THE T%0-LOOP EFFECTIVE POTENTIAL OF. . . 37S1

v' "+a'"y'=o,BM

(3.15) V"'= —a y'lnMo

where [(p ——,'A, P) —p jln

a'"= (6A, +—', g, + —,'g,g2+ —,'g2 —6g, ) . (3.16) +C(p, p, i,,g, MO), (3.26)

Integration of Eq. (3.15) yields

V" ~ = a' —"y'lnM+ C(y, p, X,g „g,,g, ), (3.17)

where

a'=a' '— 6A.

64m(3.27)

where C is a function to be determined by specifying anormalization condition on V,z. A convenient choice tofix Cis

This manner of decomposition shows that the yetunspecified function C also decomposes into

V'"=O at M=y . (3.18)C=p C, g, +(p ——'A, p ) C2

Mo

This gives +P C34 (3.28)

C =a"'y'lny .

Therefore the one-loop effective potential of the WS mod-el which is consistent with the RG functions of the modelbecomes

V ~ p2P2+ f4+ g (1)$41n(y2/M2)eff 8

where

g (1} I g(1)—2

(3.19)

(3.20)

This is to be compared with the usual result. '

It is to be noticed that the normalization condition(3.18) gives the normalization conditions

where C3 is a constant to be chosen such that V vanisheswhen P is set to zero. The two functions C, and C2 areto be fixed by specifying renormalization conditions on Vsuch as d V/dP and d V/dP at /=M. However, asimpler method than this, as dictated by Eq. (3.28), canbe applied at least for the leading-logarithmic approxima-tion. First, C& is fixed by matching V, in the limit p ~0,with the corresponding massless case. Second, C2 is thenfixed by considering the limit g~0. The resulting one-loop effective potential, in this case, now reads

2

V ff = ,P p +———p+ —,ap ln2 2 ~ 4 ] 4

eff2= —p

/=0(3.21) (p ——,'A, P ) ln

48m.

—p +2Ap

M

V,~ ——3A. ,Bf /=M =$0

where $0 is given by the minimum conditions

(3.22) 2—p'lnM

This is to be compared with Ref. 20.

(3.29)

eff (3.23) IV. THE TWO-LOOP POTENTIAL

If, however, radiative corrections due to the scalar massterm are taken into account, then Eq. (3.13) becomes

(3.24)

and Eq. (3.17) now reads

V"'= —a"'P + Ap P ln3 M

24~ Mo

In the previous section we have shown in detail how toobtain the one-loop effective potential by imposing loopexpansion on the RGE of V,~. In this section we followthe same procedure as in the previous one; however, itwill be less detailed.

Calculated to two loops, the evolution equations forthe gauge couplings are given by' ' '

(4.1)

+C(g, p, k, ,g, MO), (3.25)

where the two-loop contribution to Pg isRl

3

(4.2)

where Mo is some reference scale. Equation (3.25) can berewritten as and for P we have

3752 HAMAD ALHENDI 37

(4.3)

where3

16mP = ( g + —g2 ——g).2 (2) 3 2 25 2 3

gy 16~2 2 6 2 t

Similarly, the evolution equation for g, is given by

(4.4)

(4.5)

and the corresponding one for k is given by"

(4.6)

(4.7)

where where

16m P~' —— [—39k, +9k, (g1+ g2 —4g, ) —A( —„g)+—,g)g2+ —'„'g2+ —,g, )

(4.8)

For P 2 we have

P 2———2y, (4.9)

Therefore the two-loop effective potential of the WS mod-el becomes

where y is the two-loop anomalous dimension in the Lan-dau gauge given by

(4.10)+ Q (1)( 1+y(1))+ 7 + P(2) $4ln($2/M )

4 16

where + ( g (2)$4(]np2/M2)2 (4.17)

—5g'( —",,g1+-',g 2)) ~ (4.11)

where

y(2) g (2)y4 g(2)y41n(y2/~2)M

(4.12)

(2) P(2)g (2) 7 + ~ +2 (1)g (1)

2 8(4.13)

) 4y())Q (1)+3 + P())( 3+ 2)

(2 p',"~]28~2 256~2

a(1)( 3 3 2 3 n(1) 32) g2 g2+g)g)22vg gg

256vr

Therefore

(4.14)

(4.15)

The one-loop contributions p"', p'", f3"', )33~&", g", andy"' are as given in Sec. III.

Applying Eq.(2.9) to Eq. (3.1), and Eq. (3.19) using Eqs.(3.9)—(3.14) and Eqs. (4.1)—(4.11) one gets

As can be easily checked, this expression solves Eq. (2.9)

V. CONCLUSION

In this paper we have imposed the loop expansion onthe renormalization-group equation of the full effective

potential, and obtained a recursion formula for the calcu-lation of the nth-loop contribution. Application of themethod to the calculation of the two-loop effective poten-tial of the Weinberg-Salam model shows its direct use and

simplicity. The method ensures that the obtained expres-sion is consistent with the renormalization-group func-

tions of the model. One merit of this method is that all

quantities participating in the calculation are already re-normalized. Calculation of higher-order contributions is

straightforward. In particular, it is easy to include gaugecontributions to the third loop. The result of Eq. (4.17)can be used to investigate the vacuum stability of the WSmodel at the two-loop level. At present this point is un-

der investigation.It should be pointed out that the present method may

not be applicable to Green's functions as one does not ex-pect the RGE's and the renormalization conditions to besufficient for their determination. In fact, the solution ofthe RGE's for these functions depends on a momentumscale variable (Ref. 17).

ACKNOWLEDGMENT

where we have used the normalization condition

v("(y=m) =0 . (4.16)

The author would like to thank Professor M.O. Tahafor useful discussions on the scale limit of Green's func-tions.

37 CALCULATING THE TWO-LOOP EFFECTIVE POTENTIAL OF. . . 3753

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