Renormalization Group Improvement of the

arXiv:hep-ph/9207252v2 21 Jul 1993UCLA/92/TEP/26July 1992hep-ph/9207252Renormalization Group Improvement of theEffective Potential in Massive O(N) Symmetric φ4 TheoryBoris Kastening1Department of PhysicsUniversity of California, Los AngelesLos Angeles, California 90024AbstractThe renormalization group is used to improve the effective potential of massive O(N)symmetric φ4 theory. Explicit results are given at the two-loop level.1email after 08/26/92: boris@ruunts.fys.ruu.nl

Recently techniques have been developed [1] that allow to extend renormalizationgroup (RG) improvement of the effective potential (EP) in one-component φ4 theoryfrom the case of vanishing bare mass parameter [2] to the massive case. In this paper oneof those methods is extended to the O(N) symmetric model.

This is interesting becausethe model contains Goldstone bosons if O(N) is spontaneously broken.Also two different types of logarithms appear in the course of infinite renormalizationsince the Higgs and Goldstone boson masses depend differently on the classical back-ground field.In more realistic theories typically several such logarithms appear, e.g.through the additional presence of fermions and gauge bosons that get their massesthrough Yukawa and gauge couplings, respectively, when the Higgs field acquires anonzero vacuum expectation value. Since RG improvement at the one-, two-, .

. .

looplevel amounts to summing up leading, next-to-leading, . .

. logarithmic terms of the po-tential, one has to worry about the meaning of that statement when more than one kindof logarithm is present.Our model is defined by the LagrangianL = 12∂µφi∂µφi −V0,V0 = λ4!φ4 + m22 φ2,(1)where φ2 ≡φiφi and i = 1, .

. .

, N. The Feynman rules are easily worked out and onecan compute the one-, two-, . .

. loop contribution to the EP, e.g.

making use of vacuumgraphs in a shifted theory [3].In order to RG improve the EP, we first need the unimproved potential. With dimen-sional regularization [4] and the MS-scheme [5], a scheme used throughout this paper,1

the one-loop contribution V1 to the EP is easily seen to be given by(4π)2V1 = m4H4 ln m2Hµ2 −32!+ (N −1)m4G4 ln m2Gµ2 −32!,(2)where m2H = λ2φ2 + m2, m2G = λ6φ2 + m2, and µ is the renormalization scale.The two-loop contribution is considerably harder to determine. However, recentlythis has been achieved [6] and the result is2(4π)4V2 =18λ2φ2m2H ln2 m2Hµ2 −4 ln m2Hµ2 + 8Ω(1) + 5!+ 18λm4H ln m2Hµ2 −1!2+(N −1)( 172λ2φ2"(m2H + 2m2G) ln2 m2Gµ2 −4 ln m2Gµ2 + 8Ω m2Hm2G!+ 5!+2m2H ln m2Hµ2 ln m2Gµ2 −2!#+ 112λm2Hm2G"ln m2Hµ2 ln m2Gµ2 −ln m2Hµ2 −ln m2Gµ2 + 1#)+(N2 −1) 124m4G ln m2Gµ2 −1!2,(3)where Ωis defined byΩ(x) ≡qx(4 −x)x + 2Z arcsin( 12√x)0ln(2 sin t)dtforx ≤4qx(x −4)x + 2Z arcosh( 12√x)0ln(2 cosh t)dtforx > 4.

(4)Since (see [6]) limx→∞{Ω(x) −[ 18 ln2 x + 14ζ(2)]} = 0, V2 is finite at the tree-levelminimum (where m2G = 0), as is V1. Let us define eV ≡V/φ4, y ≡ln(m2H/µ2), andz ≡2m2/(λφ2).

Because µ appears in the n-loop contribution to the EP only in terms2As one can easily convince oneself, there is a typo in the formula given in [6]. I am grateful to TimJones for pointing this out to me.2

proportional to lnkH(m2H/µ2) lnkG(m2G/µ2), where 0 ≤kH + kG ≤n, the EP as computedloop by loop can be written aseV = λ 124 + z4+∞XL=1λL+1(4π)2LLXn=0yngLn(z),(5)where we have made use of the fact that every loop introduces another factor of λ, whenwriting eV in terms of λ, y, and z.After rewriting the renormalization group equation (RGE) µ ∂∂µ + β(λ) ∂∂λ + γm(λ)m2 ∂∂m2 −γ(λ)φ ∂∂φ!V (λ, m2, φ, µ) = 0(6)into an equation for eV in terms of λ, y, and z, it is straightforward to show that (5) withthe first few gLn determined by (2) and (3) fails to obey this RGE. As in the N = 1 case[1] the problem can be cured by introducing a suitable µ-independent tree-level constantinto the potential.

Then (5) becomeseVfull=λ 124 + z4+∞XL=1λL+1(4π)2LLXn=0yngLn(z) + z2∞Xk=1bkλk≡∞XL=0λL+1(4π)2LLXn=0yngLn(z) + z2∞Xk=1bkλk≡∞Xk=1λkfk(z, t),(7)where t ≡λy/(4π)2, the bk are to be determined by demanding consistency with theunimproved k-loop potential andfk(z, t) ≡(4π)2(1−k)∞Xn=0tngk+n−1,n(z) + bkz2. (8)The tree-level potential, represented by g00 in (7), is part of f1.3

If we write ln(m2G/µ2) as ln(m2G/m2H) + ln(m2H/µ2), then fk contains all k-th leadingpowers in ln(m2H/µ2), i.e. for every n it contains the terms proportional to lnn−k+1(m2H/µ2)of the n-loop contribution to the EP.The RGE (6) can be rewritten [1] as recursive differential equations for the functionsfk:(4π)2LXk=1(βk+1t ∂∂t −(βk+1 −αk −2γk)z ∂∂z + [(L −k + 1)βk+1 −4γk])fL−k+1−2∂fL∂t +L−1Xk=1βk+1 −2γk + zαk1 + z∂fL−k∂t= 0,(9)where the αk, βk, and γk are defined byγm ≡∞Xk=1αkλk,β ≡∞Xk=2βkλk,γ ≡∞Xk=1γkλk.

(10)The boundary conditions for the equations (9) are given by (8) at t = 0, i.e.fk(z, 0) = (4π)2(1−k)gk−1,0(z) + bkz2. (11)To fix bk we demand fk to be consistent with gk1 which in turn can be extracted fromthe k-loop contribution to the EP, Vk.

With (see e.g. [6, 7])α1 =N + 23(4π)2,β2 =N + 83(4π)2,γ1 =0,α2 = −5(N + 2)18(4π)4 ,β3 = −3N + 143(4π)4 ,γ2 = N + 236(4π)4,(12)we are ready now to compute f1 and f2.Withg00(z) = λ 124 + z4(13)4

in the boundary condition (11) we can solve (9) for the case L = 1. Upon expanding theresulting expression in t and matching the linear term withg11(z) = 116(1 + z)2 + 116(N −1)(1/3 + z)2(14)gotten from (2), we get b1 = 3N/[8(N −4)] andf1(z, t) = 1241 −N + 86t−1+ z41 −N + 86t−N+2N+8+3Nz28(N −4)1 −N + 86t−N−4N+8.

(15)Note that the φ-dependent part of f1φ4 remains finite for N = 4. As a further check of(15) one can determine the t2-term in f1 and compare it with g22 gotten from (3) andfind agreement.Next we compute f2.

We can extractg10(z) = −332(1 + z)2 + 116(N −1)(1/3 + z)2"ln 1/3 + z1 + z!−32#(16)from (2) and use it in the boundary condition (11) to solve (9) for L = 2. Upon expandingthe resulting expression in t and matching the linear term withg21(z) =(N2 −1)(1/3 + z)248"ln 1/3 + z1 + z!−1#+(N −1)"(1/3 + z)(7/3 + z)48ln 1/3 + z1 + z!− z224 + 5z36 + 13216!#−(1 + z)(5 + z)16,(17)which can be extracted from (3), we get b2 = −N(N + 8)/[4(4π)2(N −4)(N + 2)] and5

(4π)2f2(z, t) =n−N+896−N+2432 t−N2−2N−20144(N+8) ln1 −N+86 t+ 116 ln1 + z1 −N+86 t6N+8+ N−1144 ln13 + z1 −N+86 t6N+8−N+8144 ln(1 + z)o 1 −N+86 t−2+n−N+216+ (N+2)(N+3)12(N+8) t−(N+2)(N2−2N−20)24(N+8)2ln1 −N+86 t+ 18 ln1 + z1 −N+86 t6N+8+ N−124 ln13 + z1 −N+86 t6N+8−N+224 ln(1 + z)oz1 −N+86 t−2 N+5N+8+n−N(3N2+2N+40)32(N+2)(N−4) + N(N+2)(13N+44)48(N+8)(N−4) t−N(N2−2N−20)16(N+8)2ln1 −N+86 t+ 116 ln1 + z1 −N+86 t6N+8+ N−116 ln13 + z1 −N+86 t6N+8−N16 ln(1 + z)oz2 1 −N+86 t−2 N+2N+8 . (18)Again one can show that the φ-dependent part of f2φ4 remains finite for N = 4.Now instead of choosing y ≡ln(m2H/µ2) as the relevant logarithm, one could havetaken x ≡ln(m2G/µ2) as another natural choice.

With s ≡λx/(4π)2, it is easy to seethat then our result would have beeneVfull =∞Xk=1λk efk(z, s)(19)with the modified functionsef1(z, s)=f1(z, s),(20)(4π)2 ef2(z, s)=(4π)2f2(z, s) −ln 1/3 + z1 + z! ∂∂sf1(z, s),(21)6

and so on. For large fields sufficiently short of the Landau pole at t = 6/(N + 8), x ≈yand s ≈t hold and our RG improved approximations to the potential do not changemuch if we use s instead of t. However, if there is spontaneous symmetry breaking due tonegative m2, the potential changes completely around the tree-level minimum.

In fact,the second derivative of the unimproved one-loop potential diverges there as does thesecond derivative of the one-loop improved result, if we use λ and s. If we expand inpowers of λ and t, this is true starting at the two-loop level. This indicates that we shouldnot trust our result for fields around or smaller than that minimum.

Neither should onetrust the unimproved result (3) there. The reason is, of course, the presence of infrareddivergences due to Goldstone bosons.In summary, we have used the renormalization group to obtain an improved versionof the effective potential in O(N) symmetric φ4 theory.

Eqs. (15) and (18) representour results at the one- and two-loop level.

The benefit of the improvement is for largefields sufficiently short of the Landau pole, while for fields around or smaller than thetree-level minimum infrared divergences make both the unimproved and the improvedresult untrustworthy.AcknowledgementsI am grateful to Roberto Peccei for useful comments on the manuscript and to TimJones for helpful communication. This work was supported in part by the Departmentof Energy under Contract No.

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