Prompt photon - jet angular correlations at central rapidities in p+A collisions

Photon-jet azimuthal correlations in proton-nucleus collisions are a promising tool for gaining information on the gluon distribution of the nucleus in the regime of non-linear color fields. We compute such correlations from the process $g+A\to q \bar{q} \gamma$ in the rapidity regime where both the projectile and target light-cone momentum fractions are small. By integrating over the phase space of the quark which emits the photon, subject to the restriction that the photon picks up most of the transverse momentum (to pass an isolation cut), we effectively obtain a $g+A\to q \gamma$ process. For nearly back-to-back photon-jet configurations we find that it dominates over the leading order process $q+A\to q \gamma$ by two less powers of $Q_\perp / Q_S$, where $\boldsymbol{Q}_\perp$ and $Q_S$ denote the net photon-jet pair momentum and the saturation scale of the nucleus, respectively. We determine the transverse momentum dependent gluon distributions involved in $g+A\to q \gamma$ and the scale where they are evaluated. Finally, we provide analytic expressions for $\langle\cos n\phi\rangle$ moments, where $\phi$ is the angle between $\boldsymbol{Q}_\perp$ and the average photon-jet transverse momentum $\tilde{\boldsymbol{P}}_\perp$, and first qualitative estimates of their transverse momentum dependence.


I. INTRODUCTION
Angular correlations between hadrons and prompt photons in a high-energy collisions have been suggested to provide insight into the gluon fields of hadrons or nuclei in the small-x, non-linear regime [1][2][3][4][5][6]. Prompt photons are those originating from hard interactions among the initial beam partons. Those studies have mostly focussed on photon or dilepton production through the leading order q → qγ process (in the field of the target) which dominates in the fragmentation region of the projectile at forward rapidities. In the central region on the other hand particle production is dominated by processes involving soft partons from both projectile and target. In particular, it has been pointed out in ref. [7] that the production of a photon can also occur via g → qqγ. Although the amplitude squared for this process is suppressed by one additional power of α s as compared to q → qγ, on the other hand the gluon density in the projectile at small x formally is of order 1/α s times the density of quarks and so these contributions should be comparable.
Here we consider angular correlations between photon with transverse momentum k γ⊥ and a jet with transverse momentum p ⊥ . We obtain this final state by integrating over the phase space of the quark from which the photon is emitted subject to the restriction that the photon takes most of the transverse momentum. The contribution from the quark to photon fragmentation is explicitly taken into account. Hence, our process becomes g → qγ. We show that in the back-to-back correlation limit where the photon-jet transverse momentum imbalance Q ⊥ = p ⊥ + k γ⊥ is much smaller than their average transverse momentumP ⊥ = (p ⊥ − k γ⊥ )/2, and smaller than the saturation scale Q S of the nucleus, the g → qγ process in fact dominates over q → qγ by two (fewer) powers of Q ⊥ /Q S .
The q → qγ process can be written in terms of a convolution of hard factors with the dipole forward scattering amplitude [1][2][3][4]6]. g → qqγ, on the other hand, in general involves the expectation value of a correlator of four Wilson lines at small x (see below and ref. [7]). In the near back-to-back limit Q 2 ⊥ /P 2 ⊥ 1, however, this correlator can be expressed in terms of transverse momentum dependent gluon distributions [8][9][10]. Below, we determine which gluon distributions appear in photon-jet production through the g → qγ process, and at which transverse momentum scale they are evaluated. This turns out to be given by the total transverse momentum in the final state, thus the process can be used to obtain information about the gluon distributions near the non-linear "saturation" scale. Our focus here is on proton-nucleus (p+A) collisions at high energies where the saturation scale Q S of the heavy ion is expected to be semi-hard, on the order of a few GeV. We employ the "Color Glass Condensate" formalism [11] to describe the strong yet weakly coupled gluon fields of the nucleus at small x. For completeness let us mention that photon-jet production in proton-proton (p+p) collisions has been analyzed in the k T -factorization approach with transverse momentum dependent gluon distributions in refs. [12]. Refs. [13,14] considered photon-jet production in collinear factorization at NLO supplemented by a parton shower generator. Lastly, such angular correlations from qq → γg annihilation in pp collisions have been discussed in ref. [15]. Our main interest here is to show how photon-jet azimuthal correlations at small-x probe the gluon fields of the heavy ion target in the non-linear regime. From the cross section for g → qγ we can compute cos nφ angular moments, where φ denotes the angle between Q ⊥ andP ⊥ . These are non-zero for all n but increasingly suppressed as (Q ⊥ /P ⊥ ) n . The transverse momentum dependence of these moments in proton-nucleus collisions provides information about the gluon distributions of the heavy ion target. We also compute the "azimuthal harmonics" cos nΦ (n = 1 . . . 3) which are frequently studied in heavy-ion collisions, where now Φ is the angle between k γ⊥ and p ⊥ .
Photon-hadron (or jet) correlations have been studied experimentally by the PHENIX [16] and STAR collaborations [17] at the BNL-RHIC accelerator, and by CMS and ATLAS at the CERN-LHC. Thus far the experiments at RHIC have focused on p+p or nucleus-nucleus (A+A) collisions. The latter provide insight mainly into final state interactions of the jet with the hot and dense medium produced in heavy-ion collisions. Photon-jet correlations in p+p collisions have provided information on transverse momentum dependent gluon distributions albeit not in the weakly coupled, non-linear regime of QCD since the saturation momentum of a proton at RHIC energies (on average over impact parameters) is well below 1 GeV [18].
The ATLAS collaboration has measured photon-jet correlations in p+p collisions at 7 TeV [19]. For p jet ⊥ > 40 GeV and E γ ⊥ > 45 GeV the correlations near φ = π can be described reasonably well by established QCD Monte-Carlo generators. The CMS collaboration, too, has analyzed these correlations in p+p and A+A collisions but also obtained data for p+Pb collisions at 5 TeV [20]. Interestingly, for p jet ⊥ > 30 GeV and 40 GeV < k γ ⊥ < 50 GeV the correlation strength near φ = π appears to be overestimated by some Monte-Carlo models, see also Fig. 6 in ref. [14]. This may potentially be related to high gluon density effects of the kind considered here, especially if confirmed by more symmetric configurations with smaller Q ⊥ (say, by going to lower k γ ⊥ > 30 GeV). This paper is organized as follows. In Sec. II we recall the LO cross section expressed in terms of the transverse momentum imbalance Q ⊥ and the average photon-jet average transverse momentumP ⊥ . The NLO amplitude is introduced in Sec. III, and expanded in powers of gluon momenta in the followup Sec. A. In Sec. IV we obtain the cross section for g → qqγ, while in Sec. B we compute the nuclear distribution functions appearing in the NLO cross section. The g → qγ cross section and the angular moments a n are obtained in Sec. V B. Our results are summarized in Sec. VI.

II. LEADING ORDER PHOTON-JET CROSS SECTION
The leading order (LO) photon-jet cross section due to scattering of a (anti-) quark off the target is given by [2][3][4][5] dσ where k γ and p are the photon and jet momenta, respectively, while l is the momentum of the quark from the proton, see Fig. 1. Here f q(q),p (x p , Q 2 ) denote the collinear quark and the antiquark distribution functions of the proton evaluated at some hard scale Q 2 and N F (x A , k 2 ⊥ ) is the forward scattering amplitude of a fundamental dipole off the nucleus. At x A 1 and in the large-N c limit this is obtained by solving the Balitsky-Kovchegov (BK) evolution equation [21].
We can rewrite the cross section in terms of the light cone momentum fraction of the outgoing quark z q = p + /l + and the transverse momentum imbalance Q ⊥ and the average transverse momentumP ⊥ which are defined via Examples for NLO diagrams for photon production in the high-energy limit. In the left diagram the gluon from the proton emits a qq pair which then scatters on the dense nucleus. In the right diagram the gluon scatters on the nucleus and then produces a qq pair and a photon. The complete NLO contribution includes diagrams where the photon is emitted from the antiquark line, or from a virtual quark anti-quark state (not shown) in case of single-inclusive photon production.
In terms of these variables, We focus on photon-jet configurations that correspond toP ⊥ Q S Q ⊥ so that the transverse momenta of the photon and the jet are almost back to back. In this limit, due to non-linear effects the dipole forward scattering is proportional to the inverse saturation scale squared of the nucleus. Hence, the leading contribution to the LO cross section is of order (Q ⊥ /Q S ) 2 , times an overall 1/P 4 ⊥ , and is independent of the angle φ between Q ⊥ andP ⊥ . An angular dependence ∼ cos φ arises at next-to leading power (Q ⊥ /Q S ) 2 Q ⊥ ·P ⊥ /P 2 ⊥ . Below we show that the cross section for g → qγ generates an isotropic contribution at order 1 (times the common 1/P 4 ⊥ ), a cos φ angular dependence already at next-to-leading power ∼ Q ⊥ /P ⊥ , and so on.

III. NEXT TO LEADING ORDER PHOTON-JET AMPLITUDE
At the NLO order photons are produced via the g → qqγ process where either the incoming gluon or either of the quarks may scatter off the field of the target, see Fig. 2. For a photon-jet final state, one of the quarks will eventually be integrated over 1 .. The external momenta in the process are q, p and k γ for the quark, antiquark and the photon, respectively. In the amplitude we also have the transverse gluon momenta from the proton, k 1⊥ , and from the nucleus, k ⊥ and k 2⊥ − k ⊥ (there are two, because of the two Wilson lines). Here, k 2⊥ ≡ P ⊥ − k 1⊥ with P ⊥ ≡ q ⊥ + p ⊥ + k γ⊥ is the total final state transverse momentum.
The main formula for the amplitude is (see Eq. (47) in [7]) HereŨ (x ⊥ ) (U (x ⊥ )) is the Wilson line in the fundamental (adjoint) representatioñ 1 Specifically, we integrate over the phase space of the quark which emits the photon. We restrict to the region of phase space where the photon picks up most of the momentum of the parent quark so that the configuration passes a photon isolation cut, see Sec. V below with t a (T a ) the SU (N c ) generators in the fundamental (adjoint) representation. gρ a p (x ⊥ ) (gρ a A (x ⊥ )) denotes the random color charge density of the proton (nucleus) which will be averaged over after squaring the amplitude. The other factor of the coupling g in Eq. (4) arises from the g − q −q vertex.
We also introduce the following momentum labels where k + = 0, k − 1 = 0, k + 1 = P + . We also define the Lipatov vertex We then have with with Here we defined the transverse mass function M (k ⊥ ) ≡ k 2 ⊥ + m 2 . We assume that there is a certain combination of the external momenta which is the largest scale in the process (i.e. the average transverse momentumP ⊥ of photon and jet). This scale is also supposed to be much larger than the saturation scale of the proton or that of the nucleus. The transverse momenta of the gluon from the proton, k 1⊥ , and of the gluons from the nucleus, k ⊥ and |k 2⊥ − k ⊥ |, are of the order of the respective saturation scales, much smaller than the hard scale.
With this assumption we perform an expansion of the amplitude in k 1⊥ , k 2⊥ − k ⊥ and k ⊥ . To obtain the gluon distributions we must expand to first order in these momenta. The main point is that by Ward identities this expansion necessarily involves terms proportional to k i 1⊥ (k 2 − k ⊥ ) j or k i 1⊥ k j ⊥ in the amplitude. Note that by conservation of momentum this implies that the total transverse momentum P ⊥ ≡ q ⊥ + p ⊥ + k γ⊥ in the final state is much smaller than the average transverse momentumP ⊥ of the photon and jet. We shall return to this point in Sec. V B. After some algebra, the full details are given in App. A, we obtain the following main result for the amplitude: where we defined with Here R µi β , with β = 3, . . . 5 (β = 6, . . . , 8) correspond to the three terms in the first (second) line of Eq. (A10), after using the momentum definitions Eq. (6) and the collinear limit: As an aside, it is useful to check the soft photon limit of the amplitude in Eq. (14). We obtain where Thus, Eq. (18) factorizes into the soft photon emission amplitude times the amplitude for qq production. The qq production part exactly agrees with the result in Ref. [10] once the expressions in Eqs. (25) and (26) qq,i and Rq ,i =T B qq,i together with k 1 → k 2 and k 2 → k 1 in the notation of Ref. [10].

IV. NEXT TO LEADING ORDER CROSS SECTION
We can immediately derive the cross section by adapting the calculation from ref. [7]. With the replacements the amplitude in Eq. (14) has the same structure as (4) (or Eq. (47) in [7]). Then, we can deduce the cross section from Eq. (63) in [7] dσ To follow more closely the notation in [10] we rewrote the nuclear gluon field correlators from Eqs. (58)-(60) in ref. [7] in terms of the four Wilson line correlator The proton and the nucleus light-cone momentum fractions, x p and x A , are related to the final state kinematics via By partial integration we can turn powers of k i , (k 2 − k) i and k i 2 into gradients of the correlator of four Wilson lines (22). For example, The cross section can be written as The gradients of C(x ⊥ , y ⊥ , y ⊥ x ⊥ ) are parametrized in terms of six transverse momentum dependent gluon distribution functions as follows (see Eqs. (28)-(34) in [10]) x The cross section becomes where The F i and H i distributions, introduced in Eqs. (27)- (29), have known operator definitions; see e. g. [10] where a direct relationships to the distributions in the conventional transverse momentum dependent (TMD) formalism discussed in [8,9] was given. Ref. [10] also established a set of linear relationship to the various dipole and Weizsäcker-Williams gluon distributions [10], see also [22]. In addition, we provide explicit expressions in terms of the BK two-point function of A + in App. B.

V. JET-PHOTON CORRELATIONS IN THE COLLINEAR APPROXIMATION
In the following we will pick up configurations where both the photon k γ⊥ and the antiquark (we arbitrarily choose to this to be the jet in this process) momentum p ⊥ are hard, while the left-over transverse momentum of the quark is much smaller, q ⊥ k γ⊥ . The contributions where the photon is emitted within a jet can be suppressed by discarding events with a large transverse jet energy in a cone around the photon [23]. The remaining configurations are in fact dominated by collinear photon emissions from a quark line after the interaction with the nuclei. For simplicity, we will also take the collinear limit on the proton side: k 1⊥ → 0. The cross section becomes where τ coll ab,ii ≡ g µµ δ jj tr ( / q + m)R a,µij (m − / p)γ 0 R † b,µ i j γ 0 , a, b = q,q , τ coll ga,ii ≡ g µµ tr ( / q + m)R g,µ (m − / p)γ 0 R † a,µ i i γ 0 , a = q,q , τ coll ag,ii ≡ g µµ tr ( / q + m)R a,µii (m − / p)γ 0 R † g,µ γ 0 , a = q,q , Here we have defined R µi a ≡ k j 1 R µij a , a = q,q and R µi g ≡ k i 1 R µ g , see Eqs. (15) and (16). Also, we introduced the collinear gluon distribution of the proton, Note that although P ⊥ is much smaller than the hard scaleP ⊥ it is nevertheless on the order of the saturation scale of the nucleus and so the collinear approximation on the proton side should be justified [24]. The terms in the amplitude (14) containing the photon-quark collinear singularity are given by R µi 1 , R µi 3 and R µi 8 , see Eq. (16). To correctly take into account this singularity the Dirac traces (33) Here we have defined where R µi α ≡ k j 1 R µij α , α = 3, . . . , 8, and R µi α ≡ k i 1 R µ α , α = 1, 2, see Eqs. (15).

A. Integration over the photon-quark collinear singularity
We introduce the photon momentum fraction and separate out the collinear singularity in the Dirac traces The integral over the transverse phase space q ⊥ is dominated by the singularity w ⊥ ≡ q ⊥ + k γ⊥ = k γ⊥ /z. The integral is calculated by expanding the integrand around this singularity and then integrating over it. A generic integral is where in the second line we shifted the integration q ⊥ → w ⊥ . F (w ⊥ , p ⊥ ) is a function containing the hard parts that potentially depend on w ⊥ and p ⊥ as well as the distribution function that depends on P ⊥ = w ⊥ + p ⊥ . In (39) we have separated out explicitly the perturbative tail ∼ 1/P 2 ⊥ of the distribution functions. Expanding F (w ⊥ , p ⊥ ) around the collinear singularity we obtain where in the second line we evaluated the integral in d-dimensions and expanded around d = 2 − 2 . Here µ 2 is the renormalization scale. Subtracting the 1/ divergence and defining µ 2 = µ 2 MS e γ E /4π, the final formula for the integral in the MS scheme is Using (41) we calculate the q ⊥ integrals in the cross section as (42)

B. Cross section in the correlation limit
We calculate τ coll, reg ab,ii (k γ⊥ /z) assuming that the total momentum of the final state in the correlation limit, p ⊥ + k γ⊥ /z, is small. This follows from the fact that the hard factors have been expanded around k 1⊥ , k 2⊥ → 0 and from momentum conservation. We introduce new variables Below we will consider the limit where Q ⊥ P ⊥ . Since this means that both Q ⊥ and |p ⊥ + k γ⊥ /z| are small, we are considering events where the photon picks up most of the energy of its parent quark, so that z is close to unity.
The explicit expressions for the hard factors are where we see explicitly the photon splitting function (1 + (1 − z) 2 )/z. We also introduced the abbreviation In terms of Q ⊥ andP ⊥ the transverse momentum in the nuclear gluon distributions is Next, we multiply the transverse projector in Eq. (42) withP iPi from (44) (47) We took the correlation limit Q ⊥ P ⊥ in the last line but in the cross section below we will keep the full expression. The full cross section, integrated over the quark momentum q ⊥ is Here we have added the contribution to the cross section coming from the fragmentation photons proportional to the quark-photon fragmentation function That way, the cross section is independent of the arbitrary renormalization scale µ 2 . It follows from (B6) that the combination of linearly polarized gluon distributions that appears in the cross section vanishes: We now expand the cross section in powers of Q ⊥ 2 : In this expansion we have neglected the dependence of x p and x A on Q ⊥ in the proton and the nuclear distributions. In both cases the derivatives will bring energy denominators that suppress such contributions in the high energy limit. In addition, the derivative of the proton distribution over log 1/x p would be α S suppressed by virtue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [25]. The nuclear distributions evolve according to the BK equation which leads to an even stronger suppression for transverse momenta of order (or less than) the saturation scale of the nucleus.
Evaluating the derivatives of any of the F j we get and and where cos φ ≡ Q ⊥ ·P ⊥ /(Q ⊥P⊥ ). Note that F The cross section, expanded up to In the collinear approximation x p,A become

C. Angular correlations
We can define angular correlations via where φ is the angle between the photon and jet transverse momentum imbalance Q ⊥ and their average transverse momentumP ⊥ . In principle, all a n can be non-zero but are increasingly suppressed like (Q ⊥ /P ⊥ ) n . From (56) we can obtain a 1 and a 2 as follows: This is the main result of this paper. Recall that ζ ≡ k + γ p + = k γ⊥ p ⊥ e η kγ −ηp is the ratio of photon and quark energies in the final state; while z is the fractional energy picked up by the photon from its parent quark. Our result applies when 1 − z 1. Note that the gluon distributions in the expressions above are evaluated at the scale (1 − z)P ⊥ /z. Hence, these angular moments provide insight into the transverse momentum dependence of the gluon distributions of the target which is due to the fact that they involve another scale, i.e. the non-linear ("saturation") scale Q S .  [26] with detailed calculations in App. B, while leaving a numerical study of evolution effects for the future. Both a 1 (P ⊥ ) and a 2 (P ⊥ ) increase rapidly with transverse momentum and reach their maximal values approximately when (1 − z)P ⊥ /z ≈ 5Q S . This maximum may disappear if evolution effects are taken into account, similar to their effect on the dipole scattering amplitude [27]. Also, we find that a 2 is substantially smaller than a 1 due to the suppression by one additional power of Q ⊥ /P ⊥ .
In Fig. 5 we plot a 1 as a function of the photon isolation cut (1 − z)/z, which is the energy fraction carried by the final state quark collinear to the photon. The behavior of a 2 is very similar. As before we used ζ = 1, Q ⊥ /P ⊥ = 0.1, P ⊥ = 15Q S , and we show curves for two different values of the saturation scale. It is interesting to observe how reducing (1 − z)/z increases the angular correlations.
We may also compute the "angular harmonics" given by where now Φ is the angle between k γ⊥ and p ⊥ . Hence, we evaluate these angular harmonics at fixed photonjet transverse momentum imbalance Q ⊥ and average transverse momentumP ⊥ by averaging over their relative orientation. Because of the requirement that Q ⊥ P ⊥ the angle Φ is close to ±π. Up to order (Q ⊥ /P ⊥ ) 2 we have cos 3Φ = −1 + 9 4 Q ⊥ P ⊥

2
(1 − cos 2φ) . Since cos 2φ = a 2 is itself of order (Q ⊥ /P ⊥ ) 2 we have, to this order, the following predictions: These expressions provide the leading dependence of v 2 1 , v 2 2 , v 2 3 on Q ⊥ /P ⊥ . These moments are insensitive to the transverse momentum dependence of the gluon distributions F i .

VI. SUMMARY AND DISCUSSION
In this paper we have computed the photon-jet cross section corresponding to the process g → qγ in the small-x regime of p+A collisions. We focus on nearly back to back configurations with a photon-jet transverse momentum imbalance Q ⊥ = k γ⊥ + p ⊥ of much smaller magnitude than their average transverse momentumP ⊥ = (p ⊥ − k γ⊥ )/2. In this limit the cross section for the process can be expressed in terms of transverse momentum dependent gluon distributions for the heavy-ion target. We determine which gluon distributions appear in this process and find, in particular, that the linearly polarized distributions do not enter (similar to q → qγ). We also show that the contribution from the leading order q → qγ process is suppressed by two powers of Q ⊥ /Q S as compared to g → qγ if Q ⊥ < Q S is less than the saturation scale of the nucleus.
The calculation begins by considering scattering of a gluon from the projectile proton off the strong color field of the target ion, thereby producing a quark, an anti-quark, and a photon. The qγ final state is obtained by integrating over the quark from which the photon was emitted (in the collinear approximation). We find that the transverse momentum scale that appears in the gluon distributions of the target is given by the left-over transverse momentum 1 − z of the quark relative to that of the photon, z, times the hard scaleP ⊥ . Hence, configurations where the photon picks up most of the momentum of the parent quark, 1 − z 1, so that a second hadronic jet collinear to the photon is not observed (due to isolation cuts), do probe the gluon distributions of the target in the non-linear regime. Finally, we provide analytic expressions and qualitative numerical estimates for a 1 = cos φ and a 2 = cos 2φ angular moments, where φ denotes the angle between Q ⊥ andP ⊥ . We predict that a 2 a 1 due to a power suppression by a factor of Q ⊥ /P ⊥ . TheP ⊥ -dependence of these angular correlations provides insight into the transverse momentum dependence of the gluon distributions of the target. In particular, numerically large a 1 and a 2 are obtained for more restrictive photon isolation cuts, and when the transverse momentum scale in the gluon distributions is on the order of a few times the saturation scale of the heavy ion target.
We have also derived analytic estimates for the "azimuthal angular harmonics" v 2 n = cos nΦ up to n = 3 and O(Q 2 ⊥ /P 2 ⊥ ), where now Φ denotes the angle between the transverse momentum k γ⊥ of the photon and that of the jet, p ⊥ . However, up to order (Q ⊥ /P ⊥ ) 2 these v n moments are insensitive to the transverse momentum dependence of the gluon distributions.
If photon-jet angular correlations in p+A collisions can indeed be studied experimentally at high-energy colliders then a more quantitative evaluation of the angular distributions than presented in this initial study would be warranted. Most importantly, small-x evolution effects on the transverse momentum dependent gluon distributions should be incorporated. This could be done by substituting the solution of the BK equation for Γ x A (x 2 ⊥ ) into the expressions for F i (x A , k 2 ⊥ ) given in sec. B. One should also account for the Sudakov suppression which arises due to the presence of the two scales Q ⊥ and P ⊥ [6,28]. On the other hand, forP ⊥ only a few times greater than Q ⊥ (to suppress power corrections reasonably well) logP ⊥ /Q ⊥ is numerically not much greater than 1 and we would not expect a very strong effect on the angular distributions. In any case, the results presented here for the g → qγ process could be used as a starting point for such improvements. Our present analysis already suggests that photon-jet correlations in p+A should provide valuable insight into transverse momentum dependent gluon distributions in the regime of non-linear color fields.
where K 1 (x) is the modified Bessel function of the second kind. Expanding around x ⊥ Λ → 0 we find However, Eq. (B1) is not restricted to the MV model. One can include small-x evolution effects by solving the BK equation [21] for Γ(x 2 ⊥ ) which now becomes also a function of x A : Γ(x 2 ⊥ ) → Γ x A (x 2 ⊥ ). We proceed with this more general formulation and express the distribution functions in terms of Γ x A (x 2 ⊥ ). We find where Γ (n) In the MV model the F -functions become