The Renormalization of the Effective Lagrangian with Spontaneous

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The renormalization of the eﬀective Lagrangian with spontaneous symmetry breaking: the SU(2) case

Qi-Shu Yan∗ and Dong-Sheng Du † Institute of high energy physics, Chinese academy of sciences, Beijing 100039, Peoples’ Republic of China

We study the renormalization of the nonlinear effective SU(2) La- grangian up to O(p4) with spontaneous symmetry breaking. The Stueck- elberg transformation, the background field gauge, the Schwinger proper time and heat kernel method, and the covariant short distance expansion technology, guarantee the gauge covariance and incooperate the Ward in- dentities in our calculations. The renormalization group equations of the effective couplings are derived and analyzed. We find that the difference between the results gotten from the direct method and the renormalization group equation method can be quite large when the Higgs scalar is far below its decoupling limit.

I. INTRODUCTION

In our last paper [1], we discussed the renormalization of the nonlinear effective U(1) Lagrangian. We learn from the case that in the framework of effective theory we can do the renormalization of the nonrenormalizable and nonlinear interactions order by order. In this paper, we use those related conceptions and methods to the non-Abelian cases. We will study the renormalization of the nonlinear effective SU(2) Lagrangian Leff up to O(p4) and derive the renormalization group equations (RGE) of its effective couplings. We will also numerically study the solutions of these RGE, and analyze the decoupling and nondecoupling effects of the Higgs boson to those effective couplings in the efective Lagrangian Leff . We find that when the Higgs scalr is far below its decoupling limit, our results are significantly different from the results gotten by matching the full theory and

∗E-mail Address: [email protected]

†E-mail Address: [email protected]

1 effective directly at the one-loop level [2] (Hereby, we call this method the direct method, in contrast with the RGE method). The basic reason for this large difference is that the direct method ignores the contribution of the possible large tree level contributions of not too heavy Higgs, which can considerably affect the effective couplings through radiative corrections. While the RGE method has taken into account these important effects. The paper is organized as following. In the section II, we briefly introduce the renormalizable SU(2) Higgs model, and concentrate on its form in unitary gauge and the quartic divergence term. In the section III, the nonlinear effective SU(2) Lagrangian Leff up to O(p4) is obtained by integrating-out the scalar Higgs boson at the tree level. We em- phasize the importance of the quartic divergence terms. In the section VI, we perform the renormalization of the Leff up to O(p4) in the background field gauge, and by using the Schwinger proper time and heat kernel method, derive the renormalization group equations so as to sum the leading logarithm contributions of radiative corrections. Section V is devoted to study the numerical solutions of these RGEs in the Higgs scalar’s decoupling and nondecoupling limits. We end the paper with some discussions and conclusions.

II. THE RENORMALIZABLE SU(2) HIGGS MODEL

The partition functional of the renormalizable non-Abelian SU(2) Higgs model [3] (Here we have not included the gauge ﬁxing term and the ghost term.) can be expressed as

a = A φ φ† exp i [A, φ, φ†] , (1) Z D µD D S Z where the action is determined by the following Lagrangian density S

1 a aµν 2 λ 2 = W W +(Dφ)† (Dφ)+µ φ†φ (φ†φ) , (2) L −4g2 µν · − 4 and the deﬁnition of quantities in this Lagrangian is given below

a a a abc b c W = ∂µW ∂νW + f W W , (3) µν ν − µ µ ν a a Dµφ = ∂µφ iW T φ, (4) − µ φ† =(φ1∗,φ2∗) , (5) where T a are the generators of the Lie algebra of SU(2) gauge group. The spontaneous symmetry breaking is induced by the positive mass square µ2 in the Higgs potential. The vacuum expectation value of Higgs ﬁeld is given as φ = v/√2. |h i| 2 And by eating the corresponding Goldstone boson, the vector bosons W obtain their mass. The non-linear form of the Lagrangian given in Eqn. (2) is made by changing the variable φ

1 iξaT a µ2 φ = (v + ρ)U,U=exp ,v=2 , (6) √2 v ! s λ where the ﬁeld U is the Goldstone boson as prescribed by the Goldstone theorem, and the ρ is a massive scalar ﬁeld. Then it reaches

2 1 a aµν (v + ρ) 1 1 2 2 λ 4 0 = W W + (DU)† (DU)+ ∂ρ ∂ρ + µ (v + ρ) (v + ρ) . (7) L −4g2 µν 2 · 2 · 2 − 16 And the change of variables induces a determinant factor in the functional integral Z a b 1 = Wµ ρ ξ exp (i 0[W, ρ, ξ]) det 1+ ρ δ(x y) . (8) Z Z D D D S v − The determinant can be written in the exponential form, and correspondingly the La- grangian density is modiﬁed to 1 0 iδ(0)ln 1+ ρ . (9) L→L − v The determinant containing quartic divergences is indispensable and crucial to cancel exactly the quartic divergences brought into by the longitudinal part of vector boson, and is important in verifying the renormalizability of the Higgs model in the U-gauge [4].

III. THE NONLINEAR EFFECTIVE SU(2) LAGRANGIAN LEFF UP TO O(P 4)

In the nonlinear effective SU(2) Lagrangian Leff , only the Goldstone and the vector bosons are included as the effective dynamic freedom at low energy region. The Lagrangian Leff , if including all permitted operators composed by these light degrees of freedom (DOFs) and respecting the assumed Lorentz and gauge symmetries, is still renormalizable [5]. Two facts are important for the actual renormalization procedure: 1) The Wilsonian renormalization method [6] and the surface theorem [7] reveals that at the low energy region, only few operators play important parts to determine the behavior of the dynamic system at the low energy region, such a fact enables us to truncate the infinite divergence tower and to consider the renormalization of the effective Lagrangian order by order; 2) While the quartic divergence terms in the effective Lagrangian enable it possible to consistently throw away all quartic divergences. 3 The general effective SU(2) Lagrangian Leff consistent with Lorenz spacetime symmetry, SU(2) gauge symmetry, and the charge, parity, and the combined CP symmetries, can be formualted as

eff = 2 + 4 + + qd , (10) L L 2 L ··· L v µ = tr[VµV ] , (11) L2 − 2 1 a aµν µ ν = W W id tr[Wµν V V ] L4 −4g2 µν − 1 µ ν µ ν +d2tr[VµVν]tr[V V ]+d3tr[VµV ]tr[VνV ] , (12) , ··· µ µ 2 tr[VµV ] (tr[VµV ]) qd = iδ(0) e1 2 + e2 4 + , (13) L ( m1 m1 ···) where the and represent the relevant and marginal operators in the Wilsonian L2 L4 renormalization method, respectively. The operators in the and also form the L2 L4 set of complete operators up to O(p4) in the usual momentum counting rule. And the higher dimension ( irrelevant ) operators than O(p4) order are represented by the dots and omitted here. The auxiliary variable Vµ is deﬁned as

Vµ = U †DµU, (14) to simplify the representation. Due to the following relations of the SU(2) gauge group 1 tr[T aT bT cT d]= (δabδcd + δadδbc δacδbd) , (15) 8 − µ ν µ ν the terms, like tr[VµVνV V ]andtr[VµV VνV ], can be linearly composed by µ ν µ ν tr[VµVν]tr[V V ]andtr[VµV ]tr[VνV ]. And since here we do not consider the term which breaks the charge, or parity, or both symmetries, therefore, the operators in Eqn. (12) are complete and linearly independent.

The effective couplings of di form the parameter space of effective theory, and they effectively reflect the dynamics of the underlying theories and the ways of symmetry breaking. Different underlying theories and ways of symmetry breaking will fall into a special point in this effective parameter space. And the di can also be called the anomalous couplings if seeing from the renormalizable SU(2) gauge theory, they refect the deviation from the requirement of renormalizability. When the scalar Higgs is heavy and integrated out, the Higgs model given in Eq. (2) can be effectively described as a special parameter point of the effective Lagragian given in Eq. (10). At the tree level, it suffices to integrate out the Higgs scalar boson by using 4 the equation of motion of it, which expresses it into the low energy dynamic DOFs and canbeformulatedas v ρ = 2 (DU)† (DU)+ , (16) m0 · ··· 1 m2 = λv2 , (17) 0 2 where m0 is the mass of Higgs bosons. The omitted terms contain at least four covariant partials and belong to higher order operators. By substituting Eqn. (16) into Eqn. (9), at the matching scale ( which is always taken at the scalar mass µ = m0 ) the effective couplings at the tree level are determined as

v2 1 d1(m0)=0,d2(m0)=0,d3(m0)= 2 = , , (18) 2m0 lambda ··· In its decoupling limit m , all these three effective couplings vanish. If a field does 0 →∞ not participate in the process of symmetry breaking, we know it will not contribute to the anomalious couplings up to the O(p4) order and its effects to the low energy dynamics will be simply suppressed by its squared mass according to the decoupling theorem [8].

IV. THE RENORMALIZATION OF LEFF AND ITS RENORMALIZATION GROUP EQUATIONS

In the background field method (BFM) [9, 10], the number of the Feynman diagrams for the loop corrections can be greatly decreased when compared with the standard Feyn- man diagram method. Another remarkable advantage is that, in the BFM, each step of calculation is manifestly gauge covariant with reference to the background gauge field, and the Ward identities — which are important to restrain the structure of divergences — have been incorporated in the calculation. The Schwinger proper time and heat kernel method [11] by itself is the Feynman integral. Combining with the covariant short distance Taylor expansion [12] in coordinate space, the divergent structures can be directly extracted out in the explicit gauge form and the loop calculation can be simplified to a considerable degree.

A. The quadratic terms of the one-loop Lagrangian

According to the spirit of the BFM, we split the Goldstone and vector bosons into classic and quantum parts, as given below

5 W W + W,U UU, (19) → → The Stueckelberg transformation [13] combinesc W andb U into the Stueckelberg ﬁeld W s

s W = U †W U + iU †∂U, (20) and eliminates the background Goldstone from the effective Lagrangian. After finishing the loop calculation, by performing the inverse Stueckelberg transformation (expanding the W s in the W and U), the effective Lagrangian can be restored to the form expressed by its low energy DOFs. As one of the advantages of the BFM, we have the freedom to choose different gauge for the background and quantum fields, and such a freedom can help to further simplify the calculation. Then for the quantum fields, we can choose the covariant gauge fixing term as

1 a abc sb c a 2 GF = [(D W ) + cf f W W + fwsξ ] , (21) L −2g2 · · c c where cf and fws are determined by requiring the one-loop Lagrangian to have the standard form given in Eqn. (26—32), then it reads 1 c = d g2 ,f = vg2 . (22) f 2 1 ws The partition functional in the background ﬁeld gauge can be expressed as Z ren s s s s =exp i [W ] =exp i tree[W ]+iδ tree[W ]+i 1 loop[W ]+ Z S S S S − ··· s s s =exp i tree[W ]+iδ tree[W ] Wµ c¯ c ξ exp i [W,ξ,c,¯ c; W ] , (23) S S D D D D S Z where the tree eﬀective Lagrangian tree isc in the following formc L 2 v s s 1 sa sµν,a 1 abc sa sµ,b sν,c tree = W W W W + d f W W W L 2 · − 4g2 µν 1 4 µν 1 1 +d W sa W sbW sa W sb + d (W s W s)2 + 2 4 · · 3 4 · ··· W s W s (W s W s)2 +iδ(0) e1 · 2 + e2 · 4 + , (24) " m1 m1 ···#

And the corresponding counter terms δ tree are defined as L 2 v sa sa 1 sa sµν,a 1 abc sa sbµ scν δ tree = δZ 2 W W δZ 2 W W + δZd d f W W W L v 2 · − g 4g2 µν 1 1 4 µν 1 sa sb sa sb 1 s s 2 +δZd d W W W W + δZd d (W W ) + 2 2 4 · · 3 3 4 · ··· W s W s (W s W s)2 +iδ(0) δe1 · 2 + δe2 · 4 + , (25) " m1 m1 ···# 6 where the renormalization constant of the Stueckelberg field W s can always be set to 1. In the one-loop level, only the quadratic terms of quantum fields are related, and they can be cast into the following standard form

1 a µν,ab b 1 a ab b a ab b quad = W 2 W + ξ 2 ξ +¯c 2 c L 2 µ WW ν 2 ξξ cc¯ 1 (µ,ab 1 *ν,ab +cW aX cξb + ξaX W b , (26) 2 µ 2 ν µν,ab 2,ab 2 ab µν µν,ab 2 = D0c + m δ g σ c , (27) WW W − WW ab ab α,ac cb αβ,ac cd db 2ξξ = 2ξξ0 + X dα + X dα dβ , (28) ab 2,ab ab 2 ab ab 20 = d + δ m + σ + σ , (29) ξξ − 1 2,ξξ 4,ξξ ab 2,ab 2 ab 2 = D0 + m δ , (30) cc¯ − W (µ,ab (µ,ac (µα,ac (µ,ab (µ,ab (µα,ab α,cd β,db cb X = Xαβ d d + X dα + X01 + X03Z + ∂αX03Y , (31) *ν,ab *ν,ac *να,ac *ν,ab *ν,ab *να,ab α,cd β,db cb X = Xαβ D0 D0 + X Dα0 + X01 + X03Z + ∂αX03Y . (32)

s s where dµ = ∂µ iaξW µ,G,andDµ0 = ∂µ iaW W µ,G. The direction of the harpoon indicates − − (µ,ab *ν,ab the position of vector bosons, and both the X and X are defined to act on the right side. For the SU(2) effective Lagrangian, the related quantities are defined as 1 σµν ab =2iW sµν,ab + d2g4(W sµ,acW sν,cb W sac W scbgµν ) WW G 4 1 G G − G · G id g2(W sµν,ab + F sµν,ab) − 1 G G d g2(W sa W sbgµν + W scW scδab + W sbW sa) − 2 · µ ν µ ν d g2(W sc W scgµν δab +2W saW sb) , (33) − 3 · µ ν σab = a2(W s W s )ab , (34) 2,ξξ − ξ G · G ab ab 1 ac sαβ,cb sαβ,cb σ = X + i aξA (W F ) 4,ξξ 4 2 αβ G − G fSαβ,acΓcd Γedb Xα,acΓcb , (35) − ξ,α ξ,β − ξ,α α,ab α,ab αβ,ab αβ,ab αβ,ac cb X = X ∂β(S + A )+2S Γ , (36) e − f ξ,β αβ,ab αβ,ab X = fS , e e e (37) µ,ab − ( µ,ab X = Se , (38) αβ − αβ µα,ab ( µα,ab µα,ab µ,ab µ,ac β,cb X = X e X ∂βX +2S Γ gαα0 , (39) 1 − 2 − βα α0β ξ µ,ab ( µ,ab X01 = Xf01 , f f e (40) (µ,ab µ,ab µ,ac α,cd β,db µα,ac µα,ac cb X = Xf S Γ Γ (X X )Γ 03Z 03 − αβ ξ ξ − 1 − 2 ξ,α aξ µ,ac sαβ,cb sαβ,cb +fi A e (W F f) , f (41) 2 αβ G − G e 7 µα,ab ( µα,ab X03Y = X2 , (42) ν,ab − * ν,ba X = Sf , (43) αβ − αβ να,ab * να,ba να,ba µ,ba ν,ca β,cb X = X e X ∂βX +2S Γ gαα0 , (44) 2 − 1 − αβ α0β W ν,ab * ν,ba X01 = Xf01 , f f e (45) *ν,ab µ,ba ν,ca α,cd β,db µα,ca µα,ca cb X = Xf S Γ Γ (X X )Γ 03Z 03 − αβ W W − 2 − 1 W,α aW µ,ca sαβ,cb sαβ,cb fi A e (W F f ) , f (46) − 2 αβ G − G *να,ab X = Xνα,bae , (47) 03Y − 1 sa abc sb scf sab acb sc ab sab ab sab where F µν = f W µ W ν , W µ,G = if W µ ,Γξ,µ = iaξW µ,G,andΓW,µ = iaW W µ,G, 2 − − with aξ =1/2andaW =(1+d1g /2) (which can be regarded as the effective charge). To get the above form, we have normalized the vector quantum gauge field by using µν,ab µν,ab W W/g. When we take the limit di 0, the σ reaches to its usual form 2iW , → → WW G as given in the gauge theory without symmetry breaking mechanism. c c As in the U(1) case, an auxiliary dimension counting rule is introduced to extract relevant terms up to O(p4), which reads

s [W µ]a =[∂µ]a =[Dµ]a =1, [v]a =0. (48) From this rule, we know

(µ,ab *ν,ab (µ,ab *ν,ab [Xαβ ]a =[Xαβ ]a =[X01 ]a =[X01 ]a =1, (49)

(µα,ab *να,ab (µα,ab *να,ab µν ab ab αβ,ab [σWW ]a =[σ2,ξξ]a =[X ]a =[X ]a =[X ]a =[X03Y ]a =[X03Y ]a =2, (50)

(µ,ab *ν,ab α,ab ab [X ]a =[X03Z ]a =[X03Z ]a =3, [σ4,ξξ]a =4. (51) We would like to mention that this auxiliary dimension counting rule is to extract those terms with the two, three and four external ﬁelds. In the limit that all anomalous couplings (µ,ab *ν,ab µν ab ab equal to zero, only the X01 , X01 , σWW ,andσ2,ξξ do not vanish. The tilded quantities are determined from the following pre-standard form [10]

ξa2ab ξb = ξa d2,ab + δabm2 ξb + ξa(σab + Xab)ξb ξξ − 1 2,ξξ 4 a α,ab b a αβ,ab b +ξ X ∂αξ + ∂αξ X ∂βξ , f (52) (µ,ab *ν,ab a b a b Wµ X ξ = ξ Xf Wν f α a µ,ab β a µα,ab b a µα,ab b c = ∂ Wµ Xcαβ ∂ ξ + Wµ X1 ∂αξ + ∂αWµ X2 ξ a µ,ab b a µ,ab b +Wcµ Xf01 ξ + WµcX03f ξ . c f (53) c f c8 f and from the eﬀective Lagrangian given in Eqn. (10), we get

Xαβ,ab = Sαβ,ab + Aαβ,ab , d 1 Sαβ,ab = 2 W sa W sbgαβ + W sα,cW sβ,cδab + Hαβ,ab f e2 e W v · 2 d e + 3 W sc W scgµν δab + Hαβ,ab , (54) v2 · W d sαβ,ab 2d d sα,a sβ,b sα,b sβ,a Aαβ,ab = i 1 W + 3 − 2 (W W W W ) , (55) − 2v2 G 2v2 − d Xe α,ab = 1 W sαβ,abW s,ab + W s,abW sαβ,ab −2v2 G β,G β,G G d d f +i 2 W sα,cW sβ,cW sab + i 3 W sc W scW sα,ab , (56) v2 β,G v2 · G d Xab = 1 W sαβ,acF scb , (57) 4 4v2 G αβ,G µ,ab µ,ab µ,ab Xfαβ = Sαβ + Aαβ , (58)

µ,ab d1g sµ,ab sab µ sab µ fS = ie (2We gαβ W g W g ) , (59) αβ 4v G − α,G β − β,G α eµ,ab d1g sab µ sab µ A = i (W g W β,Gg ) , (60) αβ − 4v α,G β − α d g Xeµα,ab = 1 (W s,ac W s,cbgµα W sα,acW sµ,cb iW αµ,ab) 1 2v G · G − G G − G d g f 2 (W sa W sbgµα + W sα,cW sµ,cδab + W sµ,bW sα,a) − v · d g 3 (W sc W scgµαδab +2W sα,bW sµ,a) , (61) − v · d g Xµα,ab = i 1 F sµα,ab , (62) 2 2v G d g2 fXµ,ab = igv(1 + 1 )W sµ,ab , (63) 01 − 2 G d g Xfµ,ab = 1 (iW sµ,acW sα,cdW scb + iW sα,acW sµ,cdW scb + W sacW sµα,cb) , (64) 03 2v G G α,G G G α,G α,G G

fαβ,ab αβ,ab sα,a sβ,b sα,b sβ,a The HW is deﬁned as HW = W W + W W , which is symmetric on its Lorentz (group) indices. To get the above form, we have utilized the equation of motion of the vector bosons W s,whichcanbeformulatedas

2 2 2 sµν,a d1g sµν,a 2 sν,a abc d1g sb sµν,c d1g abc sb sµν,c ∂µW ∂µF = m W f (1 )W W + f W F − 2 W − − 2 µ 2 µ d g2W sν,bW sµ,aW sb d g2W sν,aW sµ,bW sb. (65) − 2 µ − 3 µ From the equation of mition given in Eqn. (65), we can get 9 2 2 sν,a 1 abc d1g sb sµν,c abc d1g sb sµν,c ∂νW = 2 ∂ν f (1 )W µ W f W µ F mW " − 2 − 2 2 sν,b sµ,a sb 2 sν,a sµ,b sb +d2g W W W µ + d3g W W W µ . (66) i sµ,a 2 6 Then we know that the (∂µW ) can only contribute to terms at most up to O(p ). sµ,a Therefore we simply set ∂µW = 0 when only considering the renormalization up to O(p4). We have also used the following relations about the Lie algbra:

f abcf cde + f adcf ceb + f aecf cbd =0, (67) to simplify the related expressions.

B. The calculation of logarithm and traces

The quadratic terms can be directly calculated in the functional integral. Then after integrating out all quantum fields, the 1 loop reads L − 1 1 loop = i [Trln 2WW + Trln 2ξξ − Zx L 2 *µ (µ 1 1 +Trln 1 X 2WW− ;µνX 2ξξ− iT r2cc¯ , (68) − − where the contribution of the ghost has a different sign due to its anticommutator relation. The Tr is to sum over the Lorentz indices, µν, group indices, ab, and the coordinate space *µ (µ 1 1 points, x. The operators in the term Trln 1 X 2− X 2− are all defined to act − WW;µν ξξ on the right side, and such a form reflects that the order of integrating-out the quantum vector boson and Goldstone fields is unphysical. The expansion of logarithm is simply expressed as 1 1 1 x ln(1 X) y = x X y x XX y x XXX y x XXXX y + ... , (69) h | − | i −h | | i−2h | | i−3h | | i−4h | | i and here the X should be understood as an operator (a matrix) which acts on the quantum states of the right side. To evaluate the trace, we will use the Schwinger proper time and heat kernel method [11] in the coordinate space. In this method, the standard propagators can be expressed as 2 1,ab ∞ dτ 2 z µν,ab x 2WW− ;µν y = d exp m1τ exp HWW(x, y; τ) , (70) h | | i 0 (4πτ) 2 − −4τ ! Z 2 1,ab ∞ dτ 2 z x 2ξξ0− y = d exp m1τ exp Hξξ,ab(x, y; τ) , (71) h | | i 0 (4πτ) 2 − −4τ ! Z 10 2 d where z = y x. The integral over the proper time τ and the factor exp [ z /(4τ)] /(4πτ) 2 − − conspire to separate the quadratic divergent part of the propagator. And the H(x, y; τ) is analytic with reference to z and τ, which means that H(x, y; τ) can be analytically expanded with reference to both z and τ.Thenwehave

H(x, y; τ)=H (x, y)+H (x, y)τ + H (x, y)τ 2 + , (72) 0 1 2 ··· where H0(x, y), H1(x, y), and, H2(x, y) are the Silly-De Witt coeﬃcients. The coeﬃcient

H0(x, y) is the pure Wilson phase factor, which indicates the phase change of a quantum statewhenmovingfromthepointx to the point y and reads

x H0(x, y)=C exp Γ(z) dz , (73) − Zy · where Γ(z) is the affine connection ( dependent on the group representation of the quantum states ) defined on the coordinate point z.AndthecoefficientC is related with the spin of the states, for vector bosons, C = gµν , and for scalar bosons, C =1. − The divergence counting rule of the integral over the coordinate space x and the proper time τ can be established as

µ [z ]d =1, [τ]d = 2 . (74) − Using Eqn. (69), the two propagators deﬁned in Eqn. (70) and Eqn. (71), and the divergence and momentum counting rule given in Eqn. (48) and (74), up to O(p4), we can get the following divergent terms

2 µν 8 1 a µν,a T¯ r ln 2WW = mW tr[gµν σWW]+ ( ΓW,µν ΓW ) Zx − 3 4 1 µµ0 ν0ν + tr[σWW( gµ0ν0 )σWW( gµν )] , (75) 2 − − 2 1 a µν,a T¯ r ln 2cc¯ = ( ΓW,µν ΓW ) , (76) Zx 3 4 2 2 2 1 a µν,a T¯ r ln 2ξξ = mW tr[σ2,ξξ]+mW tr[σ4,ξξ]+ ( Γξ,µνΓξ ) Zx 3 4 1 1 2 αβ + tr[σ2,ξξσ2,ξξ]+ mW tr[X σ2,ξξ] , (77) 2 2 *µ (µ 1 1 1 Trln 1 X 2WW− ;µνX 2ξξ− = (p4t + p3t + p2t) , (78) − ¯ 2 2 where 1/¯ = i(2/ γE +ln4π )/(16π ), γE is the Euler constant, and =4 d.The − − Γµν is the field strength tensor corresponding to the affine connection Γµ.Wehaveused the dimensional regularization scheme and the modified minimal substraction scheme to

11 extract the divergent structures in this step. The p4t represents the contributions of four * ( * ( 1 1 1 1 propagators tr(X2WW− X2ξξ0− X2WW− X2ξξ0− ), which reads

αβα β g 0 0 *µ (ν *µ0 (ν0 p4t = gµν gµ ν tr[2XαβXα β X01X01 0 0 " 4 0 0

*µ (ν *µ0 (ν0 *µ (ν *µ0 (ν0 *µ (ν *µ0 (ν0 +2 + + ] X01Xαβ Xα0β0 X01 XαβX01Xα0β0 X01 X01XαβX01Xα0β0 αβα0β0α00β00 *µ (ν *µ (ν *µ (ν *µ (ν 2 g 0 0 0 0 +mW tr[XαβXα β Xα β X01 + Xαβ Xα β X01Xα β ] . (79) 4 0 0 00 00 0 0 00 00 #

* ( 1 1 α β 1 The p3t represents the contributions of three propagators tr(X2WW− X2ξξ0− Xαβ d d 2ξξ0− ), which reads

m2 *µ (ν *µ (ν 1 *µ (ν W αβα0β0 αβ p3t = g gµν tr[X X Xα β + X X Xα β ]+ g gµν tr[X X Xαβ ] . (80) 4 01 αβ 0 0 αβ 01 0 0 2 01 01

* ( 1 1 The p2t represents the contributions of two propagrators tr(X2WW− X2ξξ0− ), which can be further divided into six groups:

p2t = tAA + tAB + tAC + tBB + tBC + tCC , (81) m2 gµν gαβα0β0δγ gαβgα0β0δγ gα0β0 gαβδγ *µ (ν W αβ α0β0 δγ tAA = ( + g g g )tr[X DδDγX ] 4 6 − 2 − 2 αβ α0β0 m2 gαβα0β0 *µ (ν *µ (ν + W g tr[ σ ] g g tr[ σµ0ν0 ] , µν XαβXα0β0 2,ξξ µµ0 νν0 Xαβ WWXα0β0 8 − 2 m g g 1 *µ (να00 *µα00 (ν W α0α00 µν αβ α0β0 αβα0β0 tAB = (g g g )tr[X Dβ X X Dβ X ] , (82) 2 − 2 αβ 0 − 0 αβ 2 αβ µ ν µ ν µ ν µ ν mW g gµν * ( * ( * ( * ( tAC = tr[X X + X X + X X + X X − 2 αβ 01 01 αβ αβ 03Z 03Z αβ *µ (να0 *µα0 (ν ∂α X X X ∂α X ] − 0 αβ 03Y − 03Y 0 αβ *µ (ν *µ (ν 1 αβ µν0ν0 g tr[gµµ gνν X σ X gµν X X σ ,ξξ] −4 0 0 αβ WW 01 − 01 αβ 2 1 1 *µ (ν *µ (ν αβα0β0 αβ α0β0 gµν ( g g g )tr[X Dα Dβ X + X Dα Dβ X ] , (83) − 6 − 4 αβ 0 0 01 01 0 0 αβ 2 µα νβ mW gµν gαβ * ( tBB = tr[X X ] , (84) − 2 αβ *µα0 (ν *µ (να0 g gαα0 gµν tBC = tr[X DβX X DβX ] , (85) − 2 01 − 01 *µ (ν *µ (ν *µ (ν *µ (να0 *µα0 (ν tCC = gµν tr[X X + X X + X X ∂α X X X ∂α X ] . (86) − 01 01 01 03Z 03Z 01 − 0 01 03Y − 03Y 0 01 where the trace is to sum over the group indices and points of coordinate space, and the covariant diﬀerentials is deﬁned as

12 * ( * ( * ( * ( XDX = X∂X + XΓW X XXΓξ , (87) * ( * ( * − ( * ( * ( XDDX = X∂∂X + XΓW ΓW X + XXΓξΓξ 2XΓW XΓξ * ( * ( * −( * ( +2XΓW ∂X 2X∂XΓξ + X∂ΓW X XX∂Γξ . (88) − − And the tensors gαβγδ and gαβγδµν are symmetric on all indices and deﬁned as

gαβγδ = gαβgγδ + gαγ gβδ + gαδgβγ , (89) gαβγδµν = gαβgγδµν + gαγgβδµν + gαδgγβµν + gαµgβγδν + gαν gβγδµ . (90)

To get the p4t, p3t,andp2t, we have used the covariant shord-distance expansion technology [12] and the integral over the proper time and coordinate space. We would like to comment on the the covariant shord-distance expansion technology: to formulate the quadratic form into the standard form can simplify the labor to extract the divergences, while the form given in [12] is not easy to use. The equivalence of these two forms can be easily proved by using the partial integral. As we have pointed out, the standard form given by us has the advantage to reflect the fact that the order of integrating out the quantum vector boson and Goldstone fields has no any dynamic significance, and is unphysical.

C. The renormalization group equations

Substituting Eqs. (33—64) to Eqs. (75—86), with somewhat tedious algebraic manip- ulation, we construct the counter terms and extract the renormalization constants. The renormalization constants yield the following RGEs

2 4 2 4 dg g 29 2 17d1 g = +6d1g , (91) dt 8π2 "− 4 − 8 # 2 2 6 dv v 3g 4 7d1 g = (5d1 +8d2 +14d3) g , (92) dt 16π2 "− 2 − − 2 #

dd1 1 1 11d1 11d2 2 = + +11d3 g dt 8π2 (−12 2 − 2 ! 2 3 6 69d1 4 23d1 g + + d1 (3d2 6d3) g + , (93) "− 8 − # 48 ) 2 dd2 1 1 119d1 2 173d1 4 = + +6d2 g + +20d1d2 g dt 8π2 (−12 − 48 ! 24 ! 3 4 8 207d1 2 91d2 5d3 6 545d1 g + + d1 g + , (94) " 16 8 − 4 !# 96 )

13 dd3 1 1 89d1 19d2 2 = + 11d3 g dt 8π2 (−24 48 − 2 − ! 2 251d1 61d3 4 + d1 28d2 + g " 48 − 2 !# 3 4 8 239d1 2 53d2 43d3 6 463d1 g + d1 + g + . (95) " 16 − 4 4 !# 96 )

About the RGEs given in Eqs. (91—95), it is remarkable that the direct method will only get part of the result of the RGE method, which is contributed by the Goldstone boson and indicated by the constant terms independent of di,i=1, 2, 3intherhsofRGEsof di,i=1, 2, 3. While the rest terms of the RGEs take into account not only the eﬀect of Goldstone boson ξ, but also that of vector bosons W and that of their mixing terms. In order to compare and contrast, we formulate the results of the direct method in the c RGE form, which read

dg2 g4 29 = , (96) 2 dt 8π − 4 dv v 3 g2 = , (97) dt 16π2 "− 2 # dd 1 1 1 = , (98) 2 dt 8π −12 dd 1 1 2 = , (99) 2 dt 8π −12 dd 1 1 3 = . (100) 2 dt 8π −24 The underlying reason to represent the contributions of Higgs in the RGE form might be related with the fact that the full theory is a renormalizable one and the divergences generated by the Higgs loop should be cancelled out exactly by those generated by the Goldstone bosons. To extract the divergent structures, we have used the following relations of the SU(2) gauge group

tr[W sµW sνW s W s ]=2W sa W sbW sa W sb , (101) G G µ,G ν,G · · tr[W sµW s W sνW s ]=W sa W sbW sa W sb +(W s W s)2 , (102) G µ,G G ν,G · · · Ha Hµν,a = W a W aµν 2f abcW saW sµ,bW sν,c µν µν − µν +(W s W s)2 W sa W sbW sa W sb , (103) · − · · a where the variable Hµν is symmetric when exchanging its Lorentz indices µ and ν,andis a sa sa deﬁned as Hµν = ∂µWν + ∂ν Wµ .

14 2 The term 11d3g in the right hand side of the RGE of d1 is quite remarkable: the coeﬀecient 11 mainly comes from the fact that the gauge bosons have 3 physical components as a vector ﬁeld, and have 3 components as an adjoint representation of the SU(2).

When the Higgs is not too heavy, the coupling d3 can reach order 0.1or0.01, then this term can switch the sign of d1(mW ) from positive to negative. Another remarkable fact is that it is the radiative corrections from the vector bosons which mostly contribute to the linear terms of di in the rhs of RGEs and dominate the running of RGEs when the nonlinear eﬀects of the RGEs is still small. In the following section we will comparatively study the solutions of these two groups of RGEs.

V. NUMERICAL ANALYSIS

We concentrate on the Higgs scalar’s effects to the effective couplings di,i=1, 2, 3. To simplify the analysis, we mimic the standard model by choosing the mass of vector boson mW to be 91 GeV. The Higgs scalar is assumed to be heavier than the vector bosons W . The initial condition for the coupling g and the vacuum expectation value v is fixed at the lower boundary point, µ = mW . The coupling g(mW ) is chosen to satisfy

g2 1 α = = , (104) g 4π 30 which gives g(mW )=0.65. and the vacuum expectation value is then ﬁxed by mW = gv, which gives v(mW ) = 140.6. While the initial condition for di,i=1, 2, 3ischosentobe

ﬁxed at the matching scale, µ = m0, as given in Eqn (18). Below we will compare the results gotten from the direct method and the RGE method. As we know, the scalar’s eﬀect includes both the decoupling mass square suppressed part as shown in Eqn. (18) and the nondecoupling logarithm part. So we consider the following three cases to trace the change of roles of these two competing parts: 1) the light scalar case, with m0 = 160 GeV, where the decoupling mass square suppressed part dominates;

2) the not too heavy scalar case, with m0 = 500 GeV, where both contributions are important; 3) the very massive scalar case, with m0 =1.2 TeV, where the nondecoupling logarithm part dominates.

The figure 1. is devoted to the first case. It is obvious that the difference of di(mW ),i=

1, 2 in two methods is quite large, and the d1(mW )’s have different signs in these two method. Figure 2. is devoted to the second case. For the d1, these two methods predict different sign with the same magnitudes. Figure 3. is devoted to the third case. And the difference between the results of these two methods is relatively small.

15 3 In the all three cases, the magnitude of the d2(mW )isabout10− in both methods. and the diﬀerence between these two methods is neglectable.

vv d1 140.6

0.1 0.2 0.3 0.4 0.5

-0.002 140.4

-0.004

140.2 -0.006

-0.008

0.1 0.2 0.3 0.4 0.5 -0.01

-0.012 139.8 (a) (b)

d2 d3 0.0006

0.398 0.0005

0.396

0.0004 0.394

0.0003 0.392

0.0002

0.1 0.2 0.3 0.4 0.5

0.0001 0.388

0.1 0.2 0.3 0.4 0.5 0.386 (c) (d) FIG. 1. The varying of v, d1, d2,andd3 with the running scale t (t =ln(m0/mW )). The matching scale is the mass of Higgs scalar, which is taken to be m0 = 160 GeV. The solid lines are the results of the RGE method, while the dashed ones are those of the direct method.

2 While the d1(mW ) can reach 10− in the RGE method, one order larger than in the direct method, when the Higgs scalar is quite small. Even when the Higgs is medium heavy, the results of these two method have the same magnitude and diﬀerent signs. Near the decoupling limit, the prediction of RGE method improves that of the direct method up to 40%—70%.

Due to its initial values at the matching scale, the d3(mW ) could have different mag- 1 2 3 nitudes in these three cases, 10− ,10− ,and10− , respectively. The differences of these two methods are small when the Higgs is relative light, and in the third case the relative difference can reach 5%—15%. The difference of the running of g is neglectable in these two methods so we have not depicted it.

From these ﬁgures, we can read out the tendency that the diﬀerence of δd1 between these two methods is larger when the Higgs scalar is further below its decoupling limit.

The underlying reason for this behavior is related with the initial value d3 at the matching 16 scale, and the related terms dependent on d3 in the RGEs given in Eq. (91—95).

vv d1 140.6

140.4 0.001

140.2

0.25 0.5 0.75 1 1.25 1.5

-0.001 139.8

139.6 -0.002

(a) (b)

d2 d3

0.00175 0.044 0.0015

0.00125 0.043

0.001 0.042 0.00075

0.0005 0.041

0.00025

0.25 0.5 0.75 1 1.25 1.5

0.25 0.5 0.75 1 1.25 1.5 (c) (d) FIG. 2. The varying of v, d1, d2,andd3 with the running scale t (t =ln(m0/mW )). The matching scale is the mass of Higgs scalar, which is taken to be m0 = 500 GeV. The solid lines are the results of the RGE method, while the dashed ones are those of the direct method.

VI. DISCUSSIONS AND CONCLUSIONS

In this paper, we have studied the renormalization of the nonlinear effective SU(2) Lagrangian with spontaneous symmetry breaking and derived its RGEs. Compared with the U(1) case, the non-Abelian case is much more complicated. And quite differently, in the SU(2) case, the gauge coupling and the anomalous couplings up to O(p4)aredriven to develop by the quantum fluctuations low energy DOFs. We also have comparatively studied the results of the direct method and the RGE method. From the numerical analysis, we see that the results of two methods are very different when the Higgs scalar is far below its decoupling limit. The underlying reason is related with the initial value of d3 at the matching scale and with the radiative correction of all low energy degrees of freedom ( both the Goldstone and vector bosons ) which contributes to the d3 terms in Eq. (91—95). Normally, when the Higgs is so light, the higher dimension operators, for instance those

17 belong to the O(p6) order, might play some considerable parts and it might be not good to use the effective theory to describe the full theory, since the Wilsonian renormalization [6] and the surface theorem given by [7] require that the low energy scale µIR is lower enough than the UV cutoff. While we see, for the medium heavy Higgs (say, m0 = 400 GeV or m0 = 800 GeV ), it is still approperiate to use it, and the difference between these two method is quite considerable for some anomalous coupling(s).

vv d1 140.6

0.0025 140.4

0.002 140.2

0.5 1 1.5 2 2.5 0.0015

139.8 0.001

139.6

0.0005 139.4

139.2 0.5 1 1.5 2 2.5 (a) (b)

d2 d3 0.0095 0.0025

0.009 0.002

0.0085 0.0015

0.008 0.001

0.0075 0.0005

0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 (c) (d) FIG. 3. The varying of v, d1, d2,andd3 with the running scale t (t =ln(m0/mW )). The matching scale is the mass of Higgs scalar, which is taken to be m0 = 1200 GeV. The solid lines are the results of the RGE method, while the dashed ones are those of the direct method.

In the RGE method, it becomes quite manifest that the effects of heavy DOF to the low energy dynamics are related with two factors, 1) the mass of the heavy particle, which determines the matching scale µUV , and 2) the initial values of anomalous couplings at the matching scale determined by integrating out the heavy particle, which are related with the spin of the heavy particle and the strength of its couplings to the low energy DOFs. If a heavy field doesn’t participate in the process of symmetry breaking, it will not contribute to the anomalious couplings up to the O(p4) order and its effects can be estimated by the decoupling theorem [8]. As we know, there are several ways for the SU(2) breaking to its subgroups, SU(2)

18 breaks to U(1) [14], for instance. In this paper, we only assume that the symmetry is broken from a local one to a global one, where all components of vector boson have the same mass. For the way of SU(2) breaking to U(1), the effective Lagrangian will be more complicated. Several of the patterns of symmetry breaking will be discussed in our next paper [16] when we consider the renormalization of electroweak chiral Lagrangian. Meanwhile, for the sake of simplicity, no fermion field is taken into account, which might introduce terms of anomaly. Also, we have not included all of terms breaking the charge, parity, and both symmetries. If included, the above procedure will be more complicated due to the properties of the complete antisymmetric tensor µνδγ . However, in principle, we can still make the renormalization procedure order by order even for the complicity. The renormalization procedure in this paper can easily be extended to study the renormalization of the nonlinear sigma model with SU(Nf ) symmetry [15], which has a very imporant role to describe the low energy hadronic physics. We will apply the related conceptions and methods to the renormalization of the electroweak chiral effective Lagrangian and the QCD chiral Lagrangian [16].

VII. ACKNOWLEDGMENTS

One of the author, Q. S. Yan, would like to thank Professor C. D. L¨u in the theory division of IHEP of CAS for helpful discussion. And special thanks to Prof. Y. P. Kuang and Prof. Q. Wang from physics department of Tsinghua university, for their kind help to ascertain some important points and to improve the representation. To manipulate the algebraic calculation and extract the β functions of RGEs, we have used the FeynCalc [17]. The work of Q. S. Yan is supported by the Chinese Postdoctoral Science Foundation and the CAS K. C. Wong Postdoctoral Research Award Foundation. The work of D. S. Du is supported by the National Natural Science Foundation of China.

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