2-loop additive mass renormalization with clover fermions and Symanzik improved gluons

Transcription

1 2-oop additive mass renormaization with cover fermions and Symanzik improved guons Apostoos Skouroupathis Department of Physics, University of Cyprus, Nicosia, CY-1678, Cyprus E-mai: Martha Constantinou Department of Physics, University of Cyprus, Nicosia, CY-1678, Cyprus E-mai: Haraambos Panagopouos Department of Physics, University of Cyprus, Nicosia, CY-1678, Cyprus E-mai: We cacuate the critica vaue of the hopping parameter, κ c, in Lattice QCD, up to two oops in perturbation theory. We empoy the Sheikhoesami-Wohert (cover improved action for Wison fermions and the Symanzik improved guon action for 4- and 6-ink oops. The quantity which we study is a typica case of a vacuum expectation vaue resuting in an additive renormaization; as such, it is characterized by a power (inear divergence in the attice spacing, and its cacuation ies at the imits of appicabiity of perturbation theory. Our resuts are poynomia in (cover parameter and cover a wide range of vaues for the Symanzik coefficients c i. Furthermore, the dependence on the number of coors N and the number of fermionic favors N f is shown expicity. In order to compare our resuts to non perturbative evauations of κ c coming from Monte Caro simuations, we empoy an improved perturbation theory method appied to improved actions. XXIV Internationa Symposium on Lattice Fied Theory Juy Tucson Arizona, US Speaker. c Copyright owned by the author(s under the terms of the Creative Commons Attribution-NonCommercia-ShareAike Licence.

2 Additive Mass Renormaization Apostoos Skouroupathis 1. Introduction In the present work, we study cover fermions [1] and Symanzik improved guons [2], for the cacuation of the critica vaue of the hopping parameter, κ c in perturbative Lattice QCD, to two oops order in a perturbative expansion. Concerning gauge fieds, we empoy the Symanzik improved action [2]. The purpose of introducing such an improvement of the attice action, is to minimize the consequences of a non-zero attice spacing, order by order in perturbation theory. Severa choices are made for the coefficients of the action corresponding to vaues which are most often used in the iterature. The discretization of the theory via attice reguarization introduces some difficuties that do not exist in the continuum theory. To be abe to recover the continuum imit, we must demand strict ocaity and absence of doubers. It is known that these requirements ead to breaking of chira symmetry. Of course, whie approaching the continuum imit we expect to recover chiraity. This is the point where one must introduce the hopping parameter. The main idea is to ensure chira symmetry, by setting the renormaized fermionic mass (m R equa to zero. Because of additive renormaization, setting the bare fermionic mass equa to zero is not enough. Hence, there is a critica mass, the roe of which is to guarantee that m R vanishes. This quantity is directy reated to the hopping parameter κ. Its critica vaue (κ c is responsibe for restoring chira symmetry. Previous works on the hopping parameter and its critica vaue appear in the iterature for Wison fermions-paquette action guons [3] and for cover fermions-paquette action guons [4]. The procedure and notation in our work is the same as in the above references. Our resuts for κ c (and thus for the critica mass depend on the number of coors (N and fermionic favors (N f. Besides that, there is an expicit dependence on the cover parameter which is kept as a free parameter. The dependence of the resuts on the choice of Symanzik coefficients cannot be given in cosed form; instead, we present it in a ist of Tabes and Figures. The rest of the paper is organized as foows: In Sec. 2 we define the actions in their discretized form, as we as the connection between the hopping parameter and fermionic mass. Furthermore, there is a description of our cacuations aong with the necessary Feynman diagrams. In Sec. 3 our resuts are presented and compared with previous Monte Caro simuations. Finay, in Sec. 4 we appy to our 1- and 2-oop resuts an improvement method, proposed by us [5, 6, 7]. This method resums a certain infinite cass of subdiagrams, to a orders in perturbation theory, eading to improved resuts. A fu write-up of this work, incuding detaied tabes of resuts is forthcoming [8]. 2. Formuation of the Probem We begin with the Wison formuation of the QCD action on the attice, with the addition of the cover (SW [1] term for fermions. In standard notation, it reads: S L = S G f f x, µ + i 4 f x (4r + m ψ f (xψ f (x [ ψ f (x ( r γ µ Uµ (xψ f (x + µ + ψ f (x + µ ( ] r + γ µ Uµ (x ψ f (x x, µ,ν ψ f (xσ µν ˆF µν (xψ f (x, (2.1 2

3 Additive Mass Renormaization Apostoos Skouroupathis where: ˆF µν 1 8a 2 (Q µν Q νµ (2.2 and: Q µν = U x,x+µ U x+µ,x+µ+ν U x+µ+ν,x+ν U x+ν,x +U x,x+ν U x+ν,x+ν µ U x+ν µ,x µ U x µ,x + U x,x µ U x µ,x µ ν U x µ ν,x ν U x ν,x +U x,x ν U x ν,x ν+µ U x ν+µ,x+µ U x+µ,x (2.3 The cover coefficient is treated here as a free parameter; r is the Wison parameter; f is a favor index; σ µν = (i/2[γ µ, γ ν ]. Powers of the attice spacing a have been omitted. Regarding guons, we use the Symanzik action, invoving Wison oops with 4 and 6 inks: S G = [c 2 g 2 0 ReTr(1 U paq + c 1 ReTr(1 U rect paq rect +c 2 ReTr(1 U chair + c 3 ReTr(1 U para chair para The correct cassica continuum imit requires: c 0 + 8c c 2 + 8c 3 = 1. The fu action is: S = S L + S g f + S gh + S m., where S g f, S gh, S m are standard gauge fixing, ghost and measure terms. The bare fermionic mass m B must be set to zero for chira invariance in the cassica continuum imit. The vaue of the parameter and of the Symanzik coefficients c i can be chosen arbitrariy; they are normay tuned in a way as to minimize (a effects. Terms proportiona to r in the action, as we as the cover terms, break chira invariance. They vanish in the cassica continuum imit; at the quantum eve, they induce nonvanishing, favor-independent corrections to the fermion masses. Numerica simuation agorithms usuay empoy the hopping parameter, κ 1 2m B a + 8r as an adjustabe quantity. Its critica vaue, at which chira symmetry is restored, is thus 1/8r cassicay, but gets shifted by quantum effects. We denote by dm the perturbative contribution that must be added to the bare mass, in order to ead to zero renormaized mass. At tree eve, m B = 0. ] (2.4 (2.5 dm = dm (1 oop + dm (2 oop (2.6 Two diagrams contribute to dm (1 oop, shown in Figure 1. The quantity dm (2 oop receives contributions from a tota of 26 diagrams, shown in Figure 2. Genuine 2-oop diagrams must be evauated at m B 0; in addition, one must incude to this order the 1-oop diagram containing an (g 2 mass counterterm (diagram 23. Certain sets of diagrams, corresponding to renormaization of 1-oop propagators, must be evauated together in order to obtain an infrared-convergent resut: These are diagrams , 12+13, , 19+20, Numerica Resuts We have seected a set of most widey used vaues for the Symanzik coefficients, shown in Tabe 1. (In a these cases, c 2 = 0. In genera, for given vaues of C 1 c 2 + c 3, C 2 c 1 c 2 c 3 the dependence on c 2 is poynomia and thus we need not choose a numerica vaue for it. 3

4 Additive Mass Renormaization Apostoos Skouroupathis Action c 0 c 1 c 3 Paquette Symanzik TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = Iwasaki DBW Tabe 1: Input parameters c 0, c 1, c Figure 1: One-oop diagrams contributing to dm (1 oop. Wavy (soid ines represent guons (fermions Figure 2: Two-oop diagrams contributing to dm (2 oop. Wavy (soid, dotted ines represent guons (fermions, ghosts. Crosses denote vertices stemming from the measure part of the action; a soid circe is a fermion mass counterterm. 4

5 Additive Mass Renormaization Apostoos Skouroupathis The contribution of the th 1-oop diagram to dm, can be expressed by: d = (N2 1 N g 2 2 i=0 c i SW ε (i (3.1 where ε (i are numerica constants and their vaues depend on the Symanzik coefficients c i. The dependence on is seen to be poynomia of degree 2 (i = 0, 1, 2. The contribution of 2-oop diagrams without cosed fermion oops takes the form d = (N2 1 N 2 g 4 c i SW N j c k (i, j,k 2 e (3.2 i, j,k where the index runs over a contributing diagrams, j = 0,2 and k = 0, 1, 2. The coefficients (i, j,k e exhibit a further dependence on Symanzik coefficients (ony through the combinations C 1, C 2, which cannot be expressed in cosed form and is presented numericay in what foows. The contribution of 2-oop diagrams, containing cosed fermion oops, has the form ẽ (i d = (N2 1 N N f g 4 4 i=0 c i SW ẽ(i (3.3 In order to enabe cross-checks, numerica per-diagram vaues of the constants ε (i (i, j,k, e and are presented in our forthcoming pubication [8], for the Iwasaki action. The tota contribution of 1-oop diagrams, for N = 3, can be written as a poynomia function of the cover parameter (. For the Wison and Iwasaki actions we find, respectivey: dm Wison (1 oop = g ( ( (1c 2 SW dm Iwasaki (1 oop = g2 ( ( ( (1c 2 SW To iustrate our 2-oop resuts for some particuar choices of the action, we set N = 3, c 2 = 0 and we use three different vaues for the favour number: N f = 0, 2. Thus, for the Wison action: N f = 0 : (3.4 (3.5 ( dm (2 oop = g ( ( (7c 2 SW (2c 3 SW (1c4 SW (3.6 N f = 2 : ( dm (2 oop = g ( ( (8c 2 SW (3c 3 SW (4c 4 SW (3.7 and for the Iwasaki action: N f = 0 : N f = 2 : ( dm (2 oop = g ( ( (4c 2 SW (2c 3 SW (1c4 SW (3.8 ( dm (2 oop = g ( ( (6c 2 SW (3c 3 SW (2c4 SW (3.9 In Figures 3 and 4 we present the vaues of dm (2 oop for N f = 0, 2, respectivey; the resuts are shown for a our choices of Symanzik actions, as a function of (N = 3, c 2 = 0. 5

6 Additive Mass Renormaization Apostoos Skouroupathis dm2-oop / g dm2-oop / g Figure 3: Tota contribution of 2-oop diagrams, for N = 3, N f = 0 and c 2 = 0. Actions (top to bottom: DBW2, Iwasaki, TILW(β =8.00, 8.10, 8.20, 8.30, 8.45, 8.60, Symanzik, Paquette. 4. Improved Perturbation Theory Figure 4: Tota contribution of 2-oop diagrams, for N = 3, N f = 2 and c 2 = 0. Actions (top to bottom: DBW2, Iwasaki, TILW(β =8.00, 8.10, 8.20, 8.30, 8.45, 8.60, Symanzik, Paquette. We now appy our method of improving perturbation theory [7], based on resummation of an infinite subset of tadpoe diagrams, termed cactus diagrams. In Ref. [7] we show how this procedure provides a simpe way of dressing (to a orders perturbative resuts at any given order (such as the 1- and 2-oop resuts of the present cacuation. Some aternative ways of improving perturbation theory have been proposed in Refs. [9, 10]. In a nutshe, our procedure invoves repacing the origina vaues of the Symanzik and cover coefficients by improved vaues, which are expicity computed in [7]. Taking aso due care to avoid doube counting of diagrams, we cacuate the improved ( dressed vaue of the critica mass (N = 3, c 2 = 0. We choose to study the case of the Wison action (β = 5.29 and the Iwasaki action (β = 1.95 with N = 3 and N f = 2. Using these vaues, the contribution to (2 oop is a poynomia of : (2 oop,wison = ( ( (2c 2 SW (8c 3 SW (1c 4 SW (4.1 (2 oop,iwasaki = ( ( (2c 2 SW (8c 3 SW (1c4 SW (4.2 The comparison between the tota dressed contribution = (1 oop +dmdr (2 oop and the unimproved contribution, dm, for the paquette action (β = 5.29, N f = 2 is exhibited in Figure 5, as a function of. Simiary, dm for the Iwasaki action (β = 1.95, N f = 2 is shown in Figure 6. Action N f β κ 1 oop κ 2 oop κ dr 1 oop κ dr 2 oop κ non pert cr Paquette Paquette Paquette Iwasaki Tabe 2: 1- and 2-oop resuts, and non-perturbative estimates for κ cr 6

7 Additive Mass Renormaization Apostoos Skouroupathis Finay, in Tabe 2, we present a comparison of dressed and undressed resuts with non perturbative estimates for κ cr [11, 12, 13]. We observe that improved perturbation theory, appied to 1-oop resuts, aready eads to a much better agreement with the non perturbative estimates dm Figure 5: Improved and unimproved vaues of dm as a function of, for the paquette action dm Figure 6: Improved and unimproved vaues of dm as a function of, for the Iwasaki action. Acknowedgements: Work supported in part by the Research Promotion Foundation of Cyprus. References [1] B. Sheikhoesami and R. Wohert, Improved continuum imit attice action for QCD with wison fermions, Nuc. Phys. B259 ( [2] K. Symanzik, Continuum imit and improved action in attice theories, Nuc. Phys. B226 ( ; Nuc. Phys. B226 ( [3] E. Foana and H. Panagopouos, The Critica Mass of Wison Fermions: A Comparison of Perturbative and Monte Caro Resuts, Phys. Rev. D63 ( [4] H. Panagopouos and Y. Proestos, The Critica Hopping Parameter in (a improved Lattice QCD, Phys. Rev. D65 ( [5] H. Panagopouos and E. Vicari, Resummation of Cactus Diagrams in Lattice QCD, Phys. Rev. D58 ( [6] H. Panagopouos and E. Vicari, Resummation of Cactus Diagrams in the Cover Improved Lattice Formuation of QCD, Phys. Rev. D59 ( [7] M. Constantinou, H. Panagopouos and A. Skouroupathis, Improved Perturbation Theory for Improved Lattice Actions, Phys. Rev. D74 ( [hep-at/ ]. [8] A. Skouroupathis, M. Constantinou and H. Panagopouos, In preparation. [9] G. Parisi, in: High-Energy Physics 1980, XX Int. Conf., Madison (1980, ed. L. Durand and L. G. Pondrom (American Institute of Physics, New York, [10] G. P. Lepage and P. B. Mackenzie, On the Viabiity of Lattice Perturbation Theory, Phys. Rev. D48 ( [11] M. Lüscher, S. Sint, R. Sommer, P. Weisz and U. Woff, Non-perturbative O(a improvement of attice QCD, Nuc. Phys. B491 ( [12] K. Jansen and R. Sommer, (a improvement of attice QCD with two favors of Wison quarks, Nuc. Phys. B530 ( [13] UKQCD Coaboration (C.R. Aton et a., Effects of non-perturbativey improved dynamica fermions in QCD at fixed attice spacing, Phys. Rev. D65 ( [hep-at/ ]. 7