We consider the abelian Higgs model with the Lagrangian:

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1 (c) 06 Roanian Journal of Physics (for accepted papers only) BOUT THE MSS CORRECTIONS IN N BELIN HIGGS MODEL RENT JOR National Institute of Physics and Nuclear Engineering PO Box MG-6, Bucharest-Magurele, Roania E-ail: rjora@theory.nipne.ro Copiled March 4, We study the corrections to the scalar boson ass in an abelian Higgs odel by considering the global properties of the partition function. We find that although the two point function for the Higgs boson receives quantu corrections, the ass of the scalar reains unaltered and thus the physical ass is equal to the bare one. 0 March 4, 06. INTRODUCTION In the context in which the Higgs boson of the standard odel has been at the forefront of both theoretical and experiental search for decades and the corrections to the Higgs boson ass or ore exactly the issues associated to the have triggered a flurry of theoretical papers it is iportant to study the probles associated with scalars and their asses in saller set-ups or ore aenable odels. The siple Φ 4 theory has been the subject of any investigations []- [3] in the last few decades, especially with regard to the behavior of the renoralized coupling constant λ R in the liit of large cut-off, the so-called triviality proble. Recent studies suggest that indeed the Φ 4 theory is trivial not only in the regie of large bare coupling constant [6], [7] but also for the full range of values of this [3]. It is iportant to analyze if the sae conclusions can be found in slightly ore coplicated theories that include scalars. In this work we extend the approach eployed in [3] to study the two point correlator and the corrections to the Higgs ass in an abelian Higgs odel with spontaneous syetry breaking. In section II we present the odel and also its counterpart in the Fourier space on a lattice. In section III we analyze the corrections to the propagator in the approach introduced in [3]. Section I is dedicated to conclusions.. THE BELIN HIGGS MODEL 9 We consider the abelian Higgs odel with the Lagrangian: RJP v.. r03b Roanian cadey Publishing House ISSN: -46X

2 Renata Jora (c) 06 RJP L = 4 F µνf µν + (D µ Φ) (D µ Φ) (Φ), () 30 where: D µ = µ + ie µ (Φ) = µ Φ Φ + λ (Φ Φ). () This odel displays spontaneous syetry breaking at the iniu of the potential: We expand the field around its vev to get: ( ) µ Φ 0 = v =. (3) λ Φ(x) = v + (h(x) + iϕ(x)), (4) where ϕ(x) is the Goldstone boson. We shall work in the R ξ gauges with ξ = where the gauge fixed Lagrangian has the for: L = µ( g µν + e v g µν ) ν + ( µh) (λv )h + ( µϕ) (ev) ϕ + e v µ ν g µν + e vh µ ν g µν + e h µ ν g µν ( µ4 λ + 8 λh4 + 8 λϕ4 + λ vh 3 + λ 4 h ϕ + λ hvϕ ) + c( + (ev) ( + h ))c. (5) v The last ter in Eq. (5) corresponds to the ghost ter of the gauge fixed Lagrangian. We shall write only the quadratic ter of the space tie integral of the La- subitted to Roanian Journal of Physics ISSN: -46X

3 (c) 06 RJP bout the ass corrections in an abelian Higgs odel 3 38 grangian L on lattice with volue in the Fourier space: d 4 xl = µ (p n )g µν (p n ) ν ( p n ) + n h(p n )(p n h )H( p n) + n ϕ(p N )(p n )ϕ( p n ) + n c(p n )(p n )c( p n ). (6) n 3. QUNTUM CORRECTIONS TO THE SCLR PROPGTOR In [3] we showed that in a siple Φ 4 theory with a real scalar field there is no correction to the all orders two point function and thus to the scalar ass, hence the Φ 4 theory is trivial. We shall apply the ethod presented there to the abelian Higgs odel. We start with the expression for the two point Higgs scalar function in the path integral foralis: Ω T h(x )h(x ) Ω = li T (+iε) dµ (x)dh(x)dϕ(x)d c(x)dc(x)h(x )h(x )exp[i d 4 xl] dµ (x)dh(x)dϕ(x)d c(x)dc(x)exp[i d 4. (7) xl] 44 We first write in the Fourier space: h(x )h(x ) = exp[ ik x ]h(k ) l exp[ ik l x ]h(k l ), (8) 45 and furtherore: Ω T h(x )h(x ) Ω = exp[ ik x ik l x ],l n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )h(k )h(k l )exp[i d 4 xl] li T (+iε) n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )exp[i d 4 (9), xl] 46 where the exponent should be expressed also in the Fourier space. subitted to Roanian Journal of Physics ISSN: -46X

4 4 Renata Jora (c) 06 RJP Next we consider the function: I x y = exp[ ik (x x )] h(k )h( k ) = exp[ ik (x x )] li T (+iε) n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )h(k )h( k )exp[id 4 xl] n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )exp[id 4 (0). xl] The relation between Eq. (9) and Eq. (0) is given by: I x y = exp[ ik (x x )] h(k )h( k ) = exp[ ik (x x )] d 4 z exp[ik z ] d 4 z exp[ ik z ] h(z )h(z ) = d 4 z d 4 z exp[ ik (x x z + z )] h(z )h(z ) = d 4 z d 4 z δ(x x z + z ) h(z )h(z ) = d 4 z h(z + x x )h(z ). () But according to the definition of the two point function the following relation holds: h(z + x x )h(z ) = d 4 p (π) 4 exp[ ip(x x )] p M(p ) = Ω T h(x )h(x ) Ω, () where M(p ) is the all order correction to the two point function in the Fourier space of the scalar h. Fro Eqs. () and () we then infer on the lattice: I x x = d 4 z Ω T h(x )h(x ) Ω = d 4 p (π) 4 exp[ ip(x x )] p M(p ), (3) where is the volue of space tie. Here it is understood that: d 4 p (π) 4 (4) in the continuu liit. This shows that in order to deterine the two point function for the Higgs boson it is sufficient to evaluate I x x. In order to do that we first separate fro the subitted to Roanian Journal of Physics ISSN: -46X

5 (c) 06 RJP bout the ass corrections in an abelian Higgs odel 5 56 Lagrangian the quadratic part as in Eq.(6). We then write: h(k )h( k ) = i δ δk n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )exp[i d 4 xl] n,p,r,s,t dµ (k n )dh(k p )dϕ(k r )d c(k s )dc(k t )exp[i d 4 xl] ϕ(k )ϕ( k ) + c(k ) c( k ) + µ (k ) µ ( k ), (5) where the quantities in the brackets are defined siilarly with the definition for the Higgs boson. One can copute the full partition function (in which no additional easure of integration is introduced) to extract the dependence on the oenta [3], + [4], [5], [6]: Z = const i ( p n n ) / i ( h p n ) i ( p n ) / i (p n ), (6) where the first factor corresponds to the Higgs boson, the second to the gauge boson, the third to the Goldstone boson and the last one to the ghosts. The Eqs. (5) and (6) lead to: i p h M(p ) = i p Σ (p ) + i p + 3 i h p i p Σ (p ) 4i p Σ (p ). (7) Here all the asses are bare asses and the quantities M(p ), Σ (p ), Σ (p ) and Σ (p ) are the corrections to the two point functions for the Higgs boson, gauge boson, Goldstone boson and ghost respectively. We can further write Eq. (7) as: 3 p + h p = p h M(p ) + p Σ (p ) p Σ (p ) + 3 p Σ (p ). (8) 68 Here in writing the right hand side of Eq. (8) we took into account two iportant 69 facts: ) The actual bare asses ξ for the Goldstone boson and the ghost is gauge 70 dependent and both the gauge paraeter and the ass can receive quantu correc tion. ) The full propagator in the Feynan gauge has apart fro the correction to the ass the sae expression as the bare one since in the odel there are no derivative interactions pertaining to the gauge boson. subitted to Roanian Journal of Physics ISSN: -46X

6 6 Renata Jora (c) 06 RJP 76 The full propagator for a boson particle X has a pole of the for: Propagator p X Σ X(p ), (9) where X is the bare ass and Σ X is the all order correction to the two point function. It is known that this propagator has a pole at the physical ass of the particle and also weaker singularities in the for of branch cuts. We expand the denoinator in the right hand side of Eq. (9) to obtain: p X Σ X (p ) = p X Σ X (p = Xphys ) Σ (p ) p = Xphys (p Xphys ) Σ (p ) p = Xphys (p Xphys )... = (p Xphys )( Σ (p ) p = Xphys Σ (p ) p = Xphys (p Xphys )...) = (p Xphys )S X(p ) Σ (p ) p = (p Xphys Xphys )..., (0) 8 where we define: X + Σ X (p = Xphys ) = Xphys ( Σ (p ) p = Xphys ) = S X = Z, () and S X ( Xphy ) = Z = (the residue of the propagator at the pole is equal to ) by 8 83 the renoralization condition. In consequence: p X Σ X(p ) = Z p + regular ters, () Xphys and there are no other poles on the right hand side of Eq. () since contributions fro two or ore particles interediate states do not produce other poles. The sae expansion can apply to all the quantities on the right hand side of Eq. (8) and leads to: 3 p + h p = (p hphys ) + (p phys ) (p phys ) + 4 (p + ters regular. (3) phys ) Then by coparing the pole structure on the right hand side and left hand side of the Eq. (3) and considering the fact that the quantities S, S, S have no zeroes we subitted to Roanian Journal of Physics ISSN: -46X

7 (c) 06 RJP bout the ass corrections in an abelian Higgs odel 7 90 conclude that: h = hphys = phys phys = phys = phys = phys = (4) Note that at first glance there are other possible cobinations of results but by considering the possible gauge dependence and the fact that the results should be correct no atter the gauge chosen and also the residue structure for this case it is clear that the possibility outlined in Eq. (4) is the only choice. final reark is in order: these results cannot be generalized easily and ay lead to different result for a nonabelian gauge theory due to the presence of the derivative ters in the Lagrangian. 4. CONCLUSIONS The renoralizability of the spontaneously broken gauge theories was treated in series of groundbreaking papers [7]- [3]. The abelian Higgs odel has been discussed in detail in [4]. It is this latter odel that we study in the present work. Based on the approach initiated in [3] we analyzed the pole structure of the derivative δz where Z is the partition function and p is the squared oentu, to find δp that the pole of the Higgs boson all order propagator is situated at the bare ass. The sae result is obtained for the assive gauge boson. However these findings do not iply that the respective propagators do not receive quantu corrections but just that these quantu corrections do not displace the poles fro the bare asses. In our study we ade the underlying assuption that the the cut-off of the theory goes to. The ethod eployed in this work cannot be extended without coplication to ore coplex odels like the non-abelian gauge theory or the standard odel due to the presence of derivative interactions and of the ferion ass ters in these theories. detailed analysis of these cases will be perfored in a separate work. The work of R. J. was supported by a grant of the Ministry of National Education, CNCS-UEFISCDI, project nuber PN-II-ID-PCE REFERENCES K. G. Wilson and J. B. Kogut, Phys. Rept., 75 (974).. M. Lindner, Zeitschrift fur Physik C 3, 95 (986). 3. I. M. Suslov, arxiv: (008). 4. D. I. Podolsky, arxiv: (00). subitted to Roanian Journal of Physics ISSN: -46X

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