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PreprinttypesetinJHEPstyle-PAPERVERSION Numeri al Study of the Ghost-Gluon Vertex in 5 Landau gauge 0 0 2 n a J 7 A. Cu hieri, T. Mendes and A. Mihara 1 Instituto de Físi a de São Carlos (cid:21) Universidade de São Paulo 2 C.P. 369, 13560(cid:21)970 São Carlos, SP (cid:21) Brazil v 4 3 0 Abstra t: We present a numeri al study of the ghost-gluon vertex and of the orrespond- 8 Z (p2) SU(2) 1 0 ing renormalization fun tion in minimal Landau gauge for latti e gauge the- = 44,84,164 4 ory. Data were obtained for tehree di(cid:27)erent latti e volumes (V ) and for three 0 β = 2.2,2.3,2.4 / latti e ouplings . Gribov- opy e(cid:27)e ts have been analyzed using the so- t a alled smeared gauge (cid:28)xing. We also onsider two di(cid:27)erent sets of momenta (orbits) in l - p order to he k for possible e(cid:27)e ts due to the breaking of rotational symmetry. The vertex (0;p,−p) e has been evaluated at the asymmetri point in momentum-subtra tion s heme. h Z (p2) p > 1 1 ∼ : We (cid:28)nd that is approximately onstant and equal to 1, at least for momenta v i GeV. This onestitutes a nonperturbative veri(cid:28) ation of the so- alled nonrenormalization of X the Landau ghost-gluon vertex. Finally, we use our data to evaluate the running oupling r α (p2) s a onstant . SU(2) Keywords: ghost-gluon vertex, latti e gauge theory, Landau gauge, running oupling onstant. Contents 1. Introdu tion 1 2. The ghost-gluon vertex 3 3. Numeri al Simulations 5 3.1 Ghost-Gluon Vertex on the Latti e 8 4. Results 10 5. Con lusions 13 1. Introdu tion In the framework of quantum (cid:28)eld theory, Faddeev-Popov ghosts are introdu ed in order to quantize non-Abelian gauge theories. Although the ghosts are a mathemati al artifa t and are absent from the physi al spe trum, one an use the ghost-gluon vertex and the ghost α (p2) s propagator to al ulate physi al observables, su h as the QCD running oupling , using the relation [1℄ Z (p2)Z2(p2) α (p2) = α 3 3 . s 0 Z2(p2) (1.1) 1 e α = g2/4π e Z (p2) Z (p2) Z (p2) 0 0 3 3 1 Here is the bare oupling onstant and , and are, respe - tively, thegluon, ghostandghost-gluonvertexrenormalizatioen fun tions.eInLandaugauge, Z (p2) 1 the vertex renormalization fun tion is (cid:28)nite and onstant, i.e. independent of the p renormalization s ale , to all orders eof perturbation theory (see [2℄ and the nonrenormalization theorems in [3, Chapter 6℄). However, a dire t nonperturbative veri(cid:28) ation of this Z (p2) 1 result isstillla king. Thisisthepurposeof thepresent work. Let usstressthat, if is (cid:28)nite and onstant also at the nonperturbative level, the equation above an be simepli(cid:28)ed, yielding a nonperturbative de(cid:28)nition of the running oupling onstant that requires only the al ulationofgluonandghostpropagators[4,5℄. Anonperturbative investigationofthe stru ture of the ghost-gluon vertex is also important for studies of gluon and ghost propagators using Dyson-S hwinger equations (DSE) [6℄. In fa t, in these studies, one makes use of Ansätze for the behavior of the propagators and verti es in the equations, in order to obtain solvable trun ation s hemes. Of ourse, a nonperturbative input for these quantities is important for a truly nonperturbative solution of the DSE. (cid:21) 1 (cid:21) Let us re all that in Landau gauge the gluon and ghost propagators an be expressed (in momentum spa e) as q q q q F(q2) Dbc(q,−q) = δbc δ − µ ν D(q2) = δbc δ − µ ν µν (cid:18) µν q2 (cid:19) (cid:18) µν q2 (cid:19) q2 (1.2) J(q2) Gbc(q,−q) = −δbc G(q2) = −δbc q2 , (1.3) F(q2) J(q2) where and are, respe tively, the gluon and ghost form fa tor and the olor b c 1,2,...,N2 − 1 SU(N ) c c indi es and take values in the ase. Then, in the momentum- subtra tion s heme one has that the gluon and ghost renormalization fun tions are given by F (q2,p2) = Z−1(p2)F(q2) R 3 (1.4) J (q2,p2) = Z−1(p2)J(q2) R 3 (1.5) e with the renormalization onditions F (p2,p2) = J (p2,p2) = 1. R R (1.6) Z (p2) = 1 1 Thus, if one an set , the running oupling (1.1) an be written as [4, 5℄ e α (p2) = α F(p2)J2(p2) , s 0 (1.7) i.e. one only needs to evaluate the gluon and ghost form fa tors de(cid:28)ned above. In re ent years, the infrared behavior of these form fa tors has been extensively studied (in Landau gauge) using di(cid:27)erent analyti al approa hes [4, 5, 7, 8, 10, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25℄. Also, numeri al studies of these form fa tors and of the running SU(2) oupling de(cid:28)ned in eq. (1.7) have been reported in [26, 27, 28, 29℄ for the group and SU(3) in [30, 31, 32, 33, 35, 36, 34, 37℄ for the ase. An indire t evaluation of the ghost-gluon vertex renormalization fun tion has been Z (p2) 1 re ently presented in [29℄, on(cid:28)rming that is (cid:28)nite in the ontinuum limit. On the other hand, a dire t nonperturbative veri(cid:28) aetion of this result from a numeri al evaluation Z (p2) 1 of the ghost-gluon vertex is still missing. Let us stress that a dire t evaluation of would allow a study of the running oupling onstant [38℄ using eq. (1.1) insteadeof eq. α (p2) s (1.7). Su h a study may improve the pre ision of the determination of , sin e in that ase one does not need, in prin iple, to use the so- alled mat hing res aling te hnique [29℄ β when onsidering data obtained at di(cid:27)erent values. Inthispaperwestudynumeri ally theghost-gluonvertex andthe orresponding renor- Z (p2) SU(2) 1 malization fun tion for the asein theminimal Landau gauge. Thede(cid:28)nition of this renormalizatieon fun tion (in the ontinuum) is presented in Se tion 2. Numeri al simulations are explained in Se tion 3. In parti ular, in Se tion 3.1 we de(cid:28)ne the vertex Z (p2) 1 renormalization fun tion on the latti e and ompare our dire t determination of to the indire t evaluation presented in [29℄. Note that, sin e the numeri al evaluatioen of the ghost-gluon vertex may be a(cid:27)e ted by Gribov- opy e(cid:27)e ts (see dis ussion in Se tions 3 and (cid:21) 2 (cid:21) q k b µ s a, c Figure 1: Notation (momenta and indi es) for the ghost-gluon vertex. 4 below), we evalute this quantity using two di(cid:27)erent gauge-(cid:28)xing methods, leading to two di(cid:27)erent setsof Gribov opies. A omparison of the results obtained in the two ases allows us to estimate the in(cid:29)uen e of Gribov opies (the so- alled Gribov noise) on the onsidered quantity. Results for the ghost-gluon vertex renormalization fun tion, the gluon and ghost prop- α (p2) s agators and the running oupling onstant are reported in Se tion 4. Finally, in Se tion 5 we draw our on lusions. Preliminary results have been reported in [39℄. 2. The ghost-gluon vertex Aa ηb µ Following the notation in Ref. [6℄ (see also Fig. 1) the 3-point fun tion for (gluon), η¯c (ghost) and (anti-ghost) (cid:28)elds (cid:22) orresponding to the ghost-gluon vertex (cid:22) is given by Vabc(x,y,z) = hAa(x)ηb(y)η¯c(z)i . µ µ (2.1) Going to momentum spa e and using translational invarian e for the 3-point fun tion, one gets Vabc(k;q,s) = d4xd4yd4z e−i(kx+qy−sz) Vabc(x,y,z) µ µ Z (2.2) = d4z e−i(k+q−s)z d4xd4y e−i(kx+qy) hAa(x)ηb(y)η¯c(0)i µ Z Z (2.3) = (2π)4 δ4(k+q−s)Gabc(k,q) , µ (2.4) δ4(k + q − s) where implies onservation of momentum. Then, the ghost-gluon vertex fun tion is obtained by (cid:16)amputating(cid:17) the orresponding 3-point fun tion (see Fig. 2) Gabc(k,q) Γabc(k,q) = µ , µ D(k2)G(q2)G(s2) (2.5) s = k +q D(k2) G(q2) where and the fun tions and have been de(cid:28)ned in eqs. (1.2) and (1.3), respe tively. At tree level (in the ontinuum) one obtains [1℄ Γabc(k,q) = ig fabcq , µ 0 µ (2.6) (cid:21) 3 (cid:21) Gabc(k,q) Γabc(k,q) µ µ Figure2: The3-pointfun tion anditsrelationwiththefullghost-gluonvertex . g fabc SU(N ) 0 c where is the bare oupling and are the stru ture fun tions of the Lie algebra. This implies the well-known result that the ghost-gluon vertex is proportional to q the momentum of the outgoing ghost. More generally, we an write the relation [6℄ Γabc(k,q) = g fabcΓ (k,q) , µ 0 µ (2.7) Γ (k,q) µ where is the so- alled redu ed vertex fun tion. Then, multiplying the previous fdbcδad a,b,c,d fdbcfabc = N δda equation by , summing over and using the relation b,c c P we get 1 Γ (k,q) = fabcΓabc(k,q) . µ g N (N2 − 1) µ (2.8) 0 c c aX,b,c Γ (k,q) = iq µ µ Sin e at tree level one has we an also write Γ (k,q) = iq Γ(k2,q2) µ µ (2.9) Γ(k2,q2) and it is the s alar fun tion that gets renormalized, when onsidering a given renormalization s heme. Clearly, from the above relations we obtain −i −i 1 Γ(k2,q2) = q Γ (k,q) = q fabcΓabc(k,q) . q2 µ µ g N (N2 − 1) q2 µ µ (2.10) Xµ 0 c c Xµ aX,b,c In our simulations we onsidered the asymmetri point with zero momentum for the k = 0 s = q gluon ( ), implying . Formula (2.10) then be omes −i 1 Ξ(q2) = Γ(0,q2) = q fabcΓabc(0,q) . g N (N2 − 1) q2 µ µ (2.11) 0 c c Xµ aX,b,c −i Let us note that the fa tor in the equations above disappears, sin e only the imaginary Γabc(0,q) Ξ(q2) µ part of the vertex fun tion ontributes to the quantity , i.e. we an write 1 1 Ξ(q2) = q fabc Γabc(0,q) . g N (N2 − 1) q2 µ Im µ (2.12) 0 c c Xµ aX,b,c (cid:21) 4 (cid:21) Figure 3: Relation between the full (left) and the bare (right) verti es. Finally, inmomentum-subtra tion s hemeone(cid:28)xesthevertexrenormalization fun tion Z (p2) 1 by requiring (see also Fig. 3) e Ξ (q2,p2) = Z (p2)Ξ(q2) R 1 (2.13) Ξ (p2,p2) = 1e, R (2.14) Γ (0,q) µ namely the renormalized redu ed vertex fun tion is equal to the tree-level value iq µ at the renormalization s ale. Then, the vertex renormalization fun tion is given by Z−1(p2) = Ξ(p2) . 1 (2.15) e We also obtain that the running oupling an be written as g (p2) = g Z−1(p2)Z1/2(p2)Z (p2) s 0 1 3 3 (2.16) = g Ξe(p2)F1/2(p2)J(pe2) 0 (2.17) 1 F1/2(p2)J(p2) = p fabc Gabc(0,p) N (N2 − 1) p2D(0)G2(p2) µ Im µ (2.18) c c Xµ aX,b,c 1 F1/2(p2) = p fabc Gabc(0,p) , N (N2 − 1) D(0)G(p2) µ Im µ (2.19) c c Xµ aX,b,c where we used eq. (1.3). 3. Numeri al Simulations For the simulations (see [40℄ for details) we onsider the standard Wilson a tion, thermal- ized using heat-bath [41℄ a elerated by hybrid overrelaxation [42, 43, 44℄. Sin e we are SU(2) U (x) µ onsidering the gauge group we parametrize the link variables as U (x) = u0(x)⊥1 + i~u (x)~σ , µ µ µ (3.1) 1⊥ 2×2 σb where is the identity matrix, are the Pauli matri es and the relation [u0(x)]2 + [~u (x)]2 = 1 µ µ (3.2) (cid:21) 5 (cid:21) V = N4 44 84 164 Latti e Volume No. of Con(cid:28)gurations 1000 400 250 Table 1: Latti e volumes and total number of on(cid:28)gurations onsidered. is satis(cid:28)ed. We also onsider the standard de(cid:28)nition of the latti e gluon (cid:28)eld, i.e. † U (x) − U (x) A (x) = Ab(x)σb = µ µ , µ µ 2i (3.3) Ab(x) ub(x) µ µ implying that the latti e gluon (cid:28)eld omponent is equal to . The onne tion U (x) A (x) µ µ between the latti e link variables and the ontinuum gauge (cid:28)eld is given by U (x) = exp iag Ab(x)tb , µ 0 µ h i (3.4) tb = σb/2 a g 0 where are the generators of the SU(2) algebra, is the latti e spa ing and 2Ab(x)/(ag ) µ 0 is the bare oupling onstant. Then, the latti e quantity approa hes the Ab(x) a → 0 µ ontinuum-gluon-(cid:28)eld omponent in the naive ontinuum limit . Let us also qˆ re all that a generi latti e momentum has omponents (in latti e units) πq a µ qˆ = 2 sin . µ (cid:18) L (cid:19) (3.5) µ e L = aN µ q µ µ µ Here is the physi al size of the latti e in the dire tion, the quantity takes ⌊−Nµ ⌋+1,...,⌊ Nµ ⌋ N µ values 2 2 and µ is the number of latti e points in the dire etion. If L q µ µ we keep the physi al size onstant and indi ate with the momentum omponents in qˆ = aq µ µ the ontinuum, we (cid:28)nd that the latti e omponents take values in the interval (−π, π] a→ 0 when . In Table 1 we show the latti e volumes used in the simulations and the orresponding numbers of on(cid:28)gurations. For ea h volume we have onsidered three values of the latti e β = 2.2,2.3,2.4 oupling, namely . The orresponding string tensions in latti e units [29℄ σa2 = 0.220(9),0.136(2) 0.071(1) are, respe tively, equal to and . Theminimal(latti e) Landau gaugeisimplemented usingthesto hasti overrelaxation (∇·A)2 algorithm [45, 46, 47℄. We stop the gauge (cid:28)xing when the average value of (cid:22) see 10−13 Eq. (6.1) in [47℄ for a de(cid:28)nition (cid:22) is smaller than . The gluon and ghost propagators in momentum spa e are evaluated using the relations 1 D(0) = Dbb(0) 12 µµ (3.6) Xb,µ 1 D(q) = Dbb(q) 9 µµ (3.7) Xb,µ e e 1 G(q) = Gbb(q), 3 (3.8) Xb e e where 1 Dbc(q) = exp[iq(x−y)]hAb(x)Ac(y)i , µν V µ ν (3.9) Xx,y e e (cid:21) 6 (cid:21) 44 84 164 V = V = V = β = % % % 2.2 4.3 25.0 97.2 β = % % % 2.3 4.4 16.8 72.8 β = % % % 2.4 4.1 9.8 50.8 Table 2: Per entage of on(cid:28)gurations for whi h the smeared gauge (cid:28)xing has found a (di(cid:27)erent) β Gribov opy. Results are reported for the three latti e volumes and the three values onsidered. Gbc(q) = h M−1 bc(q)i , (3.10) (cid:0) (cid:1) M−1 bc(qe) = 1 expe[−iq(x−y)] M−1 bc , V xy (3.11) (cid:0) (cid:1) Xx,y (cid:0) (cid:1) e e V Mbc xy is the latti e volume and is a latti e dis retization of the Faddeev-Popov operator [(−∂+A)·∂] . The expression of this matrix in terms of the gauged-(cid:28)xed link variables SU(N ) c an be found in [48, eq. (B.18)℄ for the generi ase and in [49, eq. (11)℄ for the SU(2) ase onsidered in the present work. Let us re all that this matrix has a trivial null eigenvalue orresponding to a onstant eigenve tor. Thus, one an evaluate the inverse of Mbc xy only in the spa e orthogonal to onstant ve tors, i.e. at non-zero momentum. For the inversion we used a onjugate gradient method with even/odd pre onditioning. Finally, in order to he k for possible Gribov- opy e(cid:27)e ts we have done (for ea h ther- malized on(cid:28)guration) a se ond gauge (cid:28)xing using the smearing method introdu ed in [50℄. To this end we have applied the APE smearing pro ess U (x) → (1 − w)U (x) + wΣ†(x) , µ µ µ (3.12) Σ (x) U (x) µ µ where represents the sumover the onne ting staplesforthe link , followed by w = 10/6 a reunitarization of the link matrix. We have hosen and stopped the smearing when the ondition Tr W ≥ 0.995 1,1 2 (3.13) W 1,1 was satis(cid:28)ed. (This is usually a hieved with a few APE-smearing sweeps.) Here is the 1×1 W 1,1 loop and the average is taken over all loops of a given on(cid:28)guration. The gauge (cid:28)xing for the smeared on(cid:28)guration as well as the (cid:28)nal gauge-(cid:28)xing step have been done again using the sto hasti overrelaxation algorithm. The smeared gauge-(cid:28)xing method is supposed to (cid:28)nd a unique Gribov opy, even though this opy does not always orrespond to the absolute minimum of the minimizing fun tional [50℄. In our simulations we found 44 84 164 V = V = V = β = % % % 2.2 46.5 52.0 63.8 β = % % % 2.3 38.6 52.2 65.4 β = % % % 2.4 39.0 48.7 65.4 Table 3: Per entage of (di(cid:27)erent) Gribov opies for whi h the smeared gauge (cid:28)xing has found a smallervalueoftheminimizingfun tion. Results arereportedforthethreelatti evolumesandthe β three values onsidered. (cid:21) 7 (cid:21) thatthenumber ofdi(cid:27)erent Gribov opiesobtained withthesmearedgauge(cid:28)xing in reases β with larger latti e volumes and with smaller values (see Table 2). This is in agreement with previous studies [51, 52, 53℄ and it should be related to an in reasing number of lo al minima for the minimizing fun tion when the system is highly disordered. We also found that the minimum obtained with the smeared gauge (cid:28)xing is not always smaller than the one obtained without smearing, but at larger latti e volumes the smearing approa h seems to be more e(cid:27)e tive in getting loser to the absolute minimum of the minimizing fun tion (see Table 3). 3.1 Ghost-Gluon Vertex on the Latti e On the latti e, the de(cid:28)nition of the vertex renormalization fun tion is obtained as was done in the ontinuum (see Se tion 2). The only di(cid:27)eren e is that, from the weak- oupling expansion (or (cid:16)perturbative(cid:17) expansion) on the latti e, one obtains that the ghost-gluon vertex is given at tree level by [54℄ πs a Γabc(k,q) = ig fabcqˆ cos µ , µ 0 µ (cid:18) L (cid:19) (3.14) µ e e e s = k + q a → 0 where . Clearly, by taking the formal ontinuum limit of the quantity Γabc(k,q)/a k = 0 µ e eoneere overs the ontinuum tree-level result (2.6). Then, in the ase , s = q e, eeq. (2.15) is still valid on the latti e if one sets e e e −i qˆ µ Ξ(q) = Γ (0,q) e qˆ2 Xµ cos(cid:16)πLqeµµa(cid:17) µ e (3.15) with 1 Γ (0,q) = fabcΓabc(0,q) µ g N (N2 − 1) µ (3.16) 0 c c aX,b,c e e Gabc(0,q) Γabc(0,q) = µ µ D(0)G2(q) (3.17) e e e and 1/2 4 qˆ =  qˆ2 , µ (3.18) Xµ=1   qˆ q µ µ where is de(cid:28)ned in terms of in eq. (3.5). Thus, as in the ontinuum, we an write 1 e 1 qˆ Ξ(q) = µ fabc Γabc(0,q) . e g0Nc(Nc2 − 1) qˆ2 Xµ cos(cid:16)πLqeµµa(cid:17) aX,b,c Im µ e (3.19) In order to he k for possible e(cid:27)e ts due to the breaking of rotational symmetry, in our q = q = q = 0, q = q q = 1 2 3 4 1 simulations we use two types of latti e momenta, i.e. and q = q = q = q 2 3 4 . In the following we will indi ate the (cid:28)erst tyepe ofemomentea as aesymmeetri e e e e (cid:21) 8 (cid:21) L = L = L = L = L 1 2 3 4 and the se ond one as symmetri . Clearly, using the relations , we have πqa qˆ = qˆ = 2 sin 4 (cid:18) L (cid:19) (3.20) e in the asymmetri ase and πqa qˆ = 2 sin µ (cid:18) L (cid:19) (3.21) e πqa qˆ = 4 sin (cid:18) L (cid:19) (3.22) e Ξ(q) in the symmetri one. Then, the quantity above be omes 1 1 e 1 Ξ(q) = fabc Γabc(0,q) g0Nc(Nc2 − 1) qˆ cos πLqa aX,b,c Im 4 (3.23) e (cid:16) e (cid:17) e in the asymmetri ase and 1 1 1 Ξ(q) = fabc Γabc(0,q) g0Nc(Nc2 − 1) qˆ 2 cos πLqa Xµ aX,b,c Im µ (3.24) e (cid:16) e (cid:17) e sin(2θ) = 2 sin(θ) cos(θ) in the symmetri one. Using the trigonometri relation these formulae an be rewritten as 1 Ξ(q) = Σ(q) , sin 2πqa (3.25) L e (cid:16) e (cid:17) e where 1 Σ(q) = fabc Γabc(0,q) g N (N2 − 1) Im 4 (3.26) 0 c c aX,b,c e e in the asymmetri ase and 1 1 Σ(q) = fabc Γabc(0,q) g N (N2 − 1) Im 4 µ (3.27) 0 c c aX,b,c Xµ e e in the symmetri one. Finally, using eqs. (2.3), (2.4) and (3.11) we obtain Gabc(0,q) = V hAa(0) M−1 bc(q)i , µ µ (3.28) (cid:0) (cid:1) e e where 1 Aa(0) = Aa(x) µ V µ (3.29) Xx V and is the latti e volume. Then, we an write V hAa(0) M−1 bc(q)i Γabc(0,q) = µ . µ D(0(cid:0))G2(q(cid:1)) (3.30) e e e (cid:21) 9 (cid:21)