Two-loop static potential in N=4 supersymmetric Yang-Mills theory
aa r X i v : . [ h e p - t h ] J un SFB/CPP-13-42TTP13-23
Two-loop static potential in N = 4 supersymmetricYang-Mills theory Mario Prausa and Matthias Steinhauser
Institut f¨ur Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)76128 Karlsruhe, Germany
Abstract
We compute the soft contribution to the static energy of two heavy colour sourcesinteracting via a N = 4 supersymmetric Yang-Mills theory. Both singlet and octetcolour configurations are considered. Our calculations complete recent considera-tions of the ultrasoft contributions.PACS numbers: 11.30.Pb, 12.38.Bx, 12.39.Hg, 12.60.Jv Although N = 4 supersymmetric Yang-Mills (SYM) theory is not realized in nature ithas received increasing attraction in the recent years. The main reason for this is theconjecture of a duality between N = 4 SYM and a certain class of string theories which isusually called the AdS/CFT correspondence [1]. To test the correspondence it would beideal to have all-order perturbative results which can be evaluated for large values of thecoupling and then be compared to string theory calculations. However, in general onlya few terms can be computed in the perturbative expansion and in the strong-couplinglimit. One hopes to obtain information about the AdS/CFT correspondence from theircomparison.Among the interesting quantities which one can consider there is the static energy of twoinfinitely-heavy colour sources in the fundamental representation of SU ( N c ). It has beenconsidered in the strong and weak coupling limit in Refs. [2, 3] and [4, 5], respectively. Asystematic one-loop calculation in a framework analogue to the one applied in QCD hasbeen performed in Ref. [6]. It has been noted that already at this loop-order ultrasoftcontributions have to be taken into account which is due to the massless scalar particlespresent in N = 4 SYM.ecently, in Ref. [7] the two-loop ultrasoft contribution to the static energy has beencomputed. On its own it still contains poles in ǫ which have been subtracted using theMS scheme. In this paper we provide the soft contribution to the static energy both forsinglet and octet colour configuration which combines with the ultrasoft result to a finitephysical expression for the static energy.In QCD, ultrasoft contributions arise for the first time at three-loop order [8–11] sincethe real radiation of gluons from ultrasoft quarks is suppressed by v √ α s where v is thevelocity of the heavy quark. Thus, after taking into account the scaling rule v ∼ α s thecombination of emission and absorption process scales like α s .In the next Section we provide some details to our calculation and the results are presentedin Section 3. The theoretical framework convenient for the computation of the static energy within N = 4 SYM has already been presented in Refs. [6, 7]. Let us for convenience repeat themain steps which are important for our calculation.The Lagrange density for N = 4 SYM theory reads L N =4 = − F aµν F µν a + 12 X i =1 ( D µ Φ i ) a ( D µ Φ i ) a − i X j =1 ¯Ψ aj γ µ ( D µ Ψ j ) a + . . . , (1)where Φ i ( i = 1 , . . . ,
6) represent six (pseudo) scalar particles and Ψ j ( j = 1 , . . . ,
4) fourMajorana fermions in the adjoint representation of SU ( N c ), just like the gluon fields A µ present in the field strength tensor F µν .Following [12] we introduce the Wilson loop W C = 1 N c Tr P exp (cid:20) − ig I C d τ ( A µ ˙ x µ + Φ n | ˙ x | ) (cid:21) . (2)The static energy is obtained by considering a rectangular path C and taking the limit oflarge temporal extension of the expression ( i/T ) ln h W (cid:3) i [6].The interaction with the static colour sources ψ and χ in the fundamental representationof SU ( N c ) is described via the Lagrange density L stat = ψ † ( i∂ − gA − g Φ n ) ψ + χ † c (cid:0) i∂ + gA T − g Φ Tn (cid:1) χ c . (3)Our aim is the computation of the static energy between two static sources, one in thefundamental and one in the anti-fundamental representation, to two-loop accuracy. It canbe written as E s,o = V s,o + δE US s,o , (4)2igure 1: One- and two-loop Feynman diagrams contributing to V s and V o . Thick solidlines represent heavy colour sources, thin solid lines massless fermions, curled lines gluons,and dashed lines massless scalar particles.where the subscripts “s” and “o” represent the singlet and octet representation of thesource-anti-source system, respectively. The one-loop results for V s , V o and δE US s havebeen obtained in Ref. [6] and the one- and two-loop results for δE US s and δE US o havebeen computed in Ref. [7]. In this paper we complete the next-to-next-to-leading ordercalculation and compute V s and V o to two loops. Furthermore, we add a two-loop diagramto the expression for δE US o in [7] which was omitted in that reference.We perform the calculation of V s and V o in momentum space, in close analogy to thecalculations performed in the context of QCD which are discussed in detail in the liter-ature [13–21]. The potential is obtained from the one-particle-irreducible contributionsto four-point functions with momentum exchange ~q between the static sources. SampleFeynman diagrams are shown in Fig. 1. For the singlet contribution only non-abeliancontributions have to be considered. As a consequence there are no contributions whichcontain so-called pinch-singularities of the form1( k + i k − i , (5)where k is the 0-component of the loop momentum and thus all two-loop integrals canbe reduced to one of the families shown in Fig. 2. We perform the reduction of the scalarintegrals to master integrals with the help of FIRE [22]. The analytic results for the masterintegrals are taken from Ref. [23].The colour-octet potential needs special attention since Feynman diagrams with pinchcontributions (cf. Eq. (5)) contribute to V o . The corresponding integrals cannot becomputed directly but can be reduced to integrals without pinches using the methodsdescribed in Refs. [17, 18]. In our calculation we have exploited the exponentiation of thecolour-singlet potential in order to establish relations between Feynman integrals with thesame colour factor. As an example let us consider the ladder-type diagrams which havecolour factors C F , C F C A and C F C A . Feynman diagrams with pinches are only present inthe first two cases. Exponentiation requires that the sum of all contributions proportional3igure 2: Families of scalar two-loop Feynman integrals. Solid and wavy lines representrelativistic massless and static propagators, respectively.Figure 3: Feynman diagram which contributes to δE US o . Single and double lines corre-spond to singlet and octet Greens functions, respectively. Dashed lines represent ultrasoftscalars.to C F or C F C A vanish which can be expressed through the following graphical equations + + + + + + (iterationterms) = 0 , (cid:16) + (cid:17) + 32 + + + (iterationterms) = 0 , where the Feynman diagrams represent momentum-space expressions with stripped-offcolour factors. Thick and thin lines represent static sources and massless particles (scalarsand gluons), respectively. In practice we can ignore the contributions denoted by “iter-ation terms” since they are generated by the logarithm of W C (see text below Eq. (2)).The equations can be solved for the Feynman integrals involving pinches which one inturn inserts into the expression for V o where they get multiplied by the correspondingcolour octet colour factor.We have performed our calculation in general R ξ gauge and have checked that the finalresults for V s and V o are independent of ξ which constitutes a welcome check.As stated in Eq. (4) it is necessary to add the ultrasoft contribution to the potential inorder to arrive at a finite quantity. Actually the individual contributions are divergentand contain poles in ǫ . They cancel in the sum which is a strong check both for V s,o and δE US s,o .Using the results for δE US s from Ref. [7] we indeed arrive at a finite result for E s . However,the poles do not cancel in the octet case. After examining the calculation of δE US o wehave realized that the Feynman diagram in Fig. 3 has not been considered in [7]. Only For simplicity we only consider ladder-type diagrams. E o becomes finite. For completeness we providethe result of the missing contribution which completes the list given in Appendix B of [7].Our result, which has been obtained in D = 4 − ǫ dimensions, reads(Fig. 3) = i π D g (cid:18) C A − C F (cid:19) (8 C F − C A ) ( − ∆ V ) D − Γ (cid:0) D − (cid:1) Γ(7 − D )( D − , (6)where ∆ V = V o − V s , C A = N c , C F = ( N c − / /N c , and the coupling g is defined inEq. (3). In a first step we present results for the dimensionally regularized potentials V s and V o which we parametrize in momentum space as˜ V s = − πC F α~q (cid:20) απ ˜ a (1) s + (cid:16) απ (cid:17) ˜ a (2) s + O (cid:0) α (cid:1)(cid:21) , ˜ V o = − π (cid:0) C F − C A (cid:1) α~q (cid:20) απ ˜ a (1) o + (cid:16) απ (cid:17) ˜ a (2) o + O (cid:0) α (cid:1)(cid:21) , (7)where α = g / π . After adding all contributing diagrams we obtain for the colour singletcase ˜ a (1) s = C A (cid:20) ǫ + ln (cid:18) πµ e γ ~q (cid:19) + ǫ (cid:18)
12 ln (cid:18) πµ e γ ~q (cid:19) − π (cid:19)(cid:21) + O ( ǫ ) , ˜ a (2) s = C A ( ǫ + (cid:20)
12 + π (cid:18) πµ e γ ~q (cid:19)(cid:21) ǫ − − π
12 + 12 ζ (3) + (cid:18) π (cid:19) ln (cid:18) πµ e γ ~q (cid:19) + ln (cid:18) πµ e γ ~q (cid:19) ) + O ( ǫ ) , (8)where γ ≈ . . . . is the EulerMascheroni constant. The results for the colour-octetcase read ˜ a (1) o = ˜ a (1) s , ˜ a (2) o = ˜ a (2) s + δ ˜ a (2) o ,δ ˜ a (2) o = − C A π (cid:20) ǫ + ln (cid:18) πµ e γ ~q (cid:19)(cid:21) + O ( ǫ ) . (9)One observes that, as for QCD, the one-loop results agree up to the change of the globalcolour factor from C F to C F − C A / C A π . However, in contrast to QCD, this term only contains a pole in ǫ and the corresponding logarithm. 5n coordinate space we introduce V s and V o as V s = − C F αr (cid:20) a (0) s + απ a (1) s + (cid:16) απ (cid:17) a (2) s + O (cid:0) α (cid:1)(cid:21) ,V o = − (cid:0) C F − C A (cid:1) αr (cid:20) a (0) o + απ a (1) o + (cid:16) απ (cid:17) a (2) o + O (cid:0) α (cid:1)(cid:21) , (10)and obtain for the coefficients a (0) s = 1 + ǫ ln (cid:0) πµ e γ r (cid:1) + ǫ (cid:18) π (cid:0) πµ e γ r (cid:1)(cid:19) + O (cid:0) ǫ (cid:1) ,a (1) s = C A (cid:20) ǫ + 2 ln (cid:0) πµ e γ r (cid:1) + ǫ (cid:18) π (cid:0) πµ e γ r (cid:1)(cid:19)(cid:21) + O ( ǫ ) ,a (2) s = C A (cid:20) ǫ + (cid:18)
12 + π (cid:0) πµ e γ r (cid:1)(cid:19) ǫ − π ζ (3) + (cid:18)
32 + π (cid:19) ln (cid:0) πµ e γ r (cid:1) + 94 ln (cid:0) πµ e γ r (cid:1) (cid:21) + O ( ǫ ) ,a (0) o = a (0) s ,a (1) o = a (1) s ,a (2) o = a (2) s + δa (2) o ,δa (2) o = − C A π (cid:20) ǫ + 32 ln (cid:0) πµ e γ r (cid:1)(cid:21) + O ( ǫ ) . (11)The O ( ǫ ) terms in a (0) s and a (0) o arise due to the D -dimensional Fourier transformationand are needed when inserting ∆ V into the ultrasoft expression.Note that there is no counterterm contribution since the beta function vanishes in N = 4SYM. The poles in Eqs. (8), (9) and (11) are of infra-red type and cancel against theultraviolet poles of the ultrasoft contribution. Within dimensional regularization such anidentification is necessary since scaleless integrals are set to zero (see, e.g., Ref. [24]).For completeness we also display the ultrasoft result in coordinate space which is givenby [6, 7] δE US s = − C F αr (cid:20) απ b (1) s + (cid:16) απ (cid:17) b (2) s + O (cid:0) α (cid:1)(cid:21) ,δE US o = − (cid:0) C F − C A (cid:1) αr (cid:20) απ b (1) o + (cid:16) απ (cid:17) b (2) o + O (cid:0) α (cid:1)(cid:21) , (12)6ith b (1) s = C A (cid:20) − ǫ − C A αe γ ) − (cid:0) πµ e γ r (cid:1)(cid:21) ,b (2) s = C A (cid:20) − ǫ − (cid:18)
12 + π (cid:0) πµ e γ r (cid:1)(cid:19) ǫ − (cid:18)
32 + π (cid:19) ln (cid:0) πµ e γ r (cid:1) −
94 ln (cid:0) πµ e γ r (cid:1) + (cid:18) π (cid:19) ln (2 C A αe γ ) + 2 ln (2 C A αe γ ) − − π
24 + 4 ζ (3) (cid:21) − C F C A π ,b (1) o = b (1) s ,b (2) o = b (2) s + δb (2) o ,δb (2) o = C A π (cid:20) ǫ − C A αe γ ) + 32 ln (cid:0) πµ e γ r (cid:1)(cid:21) . (13)As compared to the result given in [7] there is a change in b (2) o which is due to the missingFeynman diagram discussed in the previous Section. Note that initially the ultrasoftcontribution depends on ∆ V (cf. Eq. (6)) since V s and V o are present in the ultrasoftsinglet and octet propagators, respectively, see Appendix A of Ref. [7]. In order to arriveat Eqs. (13) the perturbative expansion of ∆ V has been inserted and an expansion in α has been performed.The comparison of the results in Eqs. (10) and (12) shows that in the sum the pole partsand the dependence on µr cancels and we arrive at the following results for the staticenergies E s = − C F αr (cid:20) απ e (1) s + (cid:16) απ (cid:17) e (2) s + O (cid:0) α (cid:1)(cid:21) ,E o = − (cid:0) C F − C A (cid:1) αr (cid:20) απ e (1) o + (cid:16) απ (cid:17) e (2) o + O (cid:0) α (cid:1)(cid:21) , (14)with e (1) s = 2 C A [ln (2 C A αe γ ) − ,e (2) s = 2 C A (cid:20) ln (2 C A αe γ ) + (cid:18) π (cid:19) ln (2 C A αe γ ) + π −
72 + 94 ζ (3) (cid:21) − C A C F π ,e (1) o = e (1) s ,e (2) o = e (2) s + δe (2) o ,δe (2) o = − C A π ln (2 C A αe γ ) . (15)The result for e (1) s and the quadratic logarithm in e (2) s agree with Ref. [6] and the linearlogarithm in e (2) s coincides with [7]. It is interesting to note that the two-loop singlet and7ctet coefficients only differ by a term proportional to π multiplied by a logarithm whichoriginates from the ultrasoft contribution in Eq. (13).In the expressions for e (1) o and e (2) o as presented above only the real part has been consid-ered. As discussed in Ref. [7] there is a nonzero imaginary part in the ultrasoft contributionwhich can be interpreted as the decay rate of the octet state into the singlet state andmassless particles. The result given in [7] changes due to the additional Feynman diagramof Fig. 3 (cf. Eq. (6)). The corrected expression for Γ o = − E o ( r )] readsΓ o = − α r C A (cid:18) C F − C A (cid:19) (cid:26) απ C A (cid:20) αC A e γ ) + 1 − π (cid:21)(cid:27) . The results discussed so far are obtained from a Wilson loop containing both the couplingto the vector bosons and scalar fields of N = 4 SYM. Alternatively, it is also possibleto consider the “ordinary” Wilson loop which is obtained by nullifying the term Φ n | ˙ x | inEq. (2), i.e. there is no interaction of the static sources and the scalars. In analogy toEq. (10) we write¯ V s = − C F αr (cid:20) απ ¯ a (1) s + (cid:16) απ (cid:17) ¯ a (2) s + O (cid:0) α (cid:1)(cid:21) , ¯ V o = − (cid:0) C F − C A (cid:1) αr (cid:20) απ ¯ a (1) o + (cid:16) απ (cid:17) ¯ a (2) o + O (cid:0) α (cid:1)(cid:21) . (16)The results for the coefficients read¯ a (1) s = − C A , ¯ a (2) s = C A (cid:20)
54 + π − π (cid:21) , ¯ a (1) o = ¯ a (1) s , ¯ a (2) o = ¯ a (2) s + δ ¯ a (2) o ,δ ¯ a (2) o = − C A (cid:20) π − π (cid:21) . (17)As expected, these results are free from ultrasoft effects which contribute starting fromthree loops. The expression for ¯ a (1) s agrees with the literature [6, 25], the other threeresults are new.To conclude, we have computed two-loop corrections to the static potential in N = 4SYM. Together with previously computed ultrasoft contributions two-loop expressionsfor the energy of two static sources are obtained where we consider the latter both in acolour-singlet and colour-octet configuration. Results are presented both for the ordinaryWilson loop and the one involving only the interaction of the static sources and the vectorfield. There seems to be a misprint in the explicit result for ¯ a (1) s as given below Eq. (31) of Ref. [6] sinceagreement with Ref. [25] is claimed. However, in [25] a different sign between the tree-level and one-loopresult is obtained. cknowledgements We would like to thank Alexander Penin and Antonio Pineda for carefully reading themanuscript and for useful comments. M.S. thanks Lance Dixon for drawing the attentionto the topic of this paper. This work was supported by DFG through SFB/TR 9.
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