Constraining Astrophysical Neutrino Flavor Composition from Leptonic Unitarity

The recent IceCube observation of ultra-high-energy astrophysical neutrinos has begun the era of neutrino astronomy. In this work, using the unitarity of leptonic mixing matrix, we derive nontrivial unitarity constraints on the flavor composition of astrophysical neutrinos detected by IceCube. Applying leptonic unitarity triangles, we deduce these unitarity bounds from geometrical conditions, such as triangular inequalities. These new bounds generally hold for three flavor neutrinos, and are independent of any experimental input or the pattern of leptonic mixing. We apply our unitarity bounds to derive general constraints on the flavor compositions for three types of astrophysical neutrino sources (and their general mixture), and compare them with the IceCube measurements. Furthermore, we prove that for any sources without $\nu_\tau$ neutrinos, a detected $\nu_\mu$ flux ratio $<1/4$ will require the initial flavor composition with more $\nu_e$ neutrinos than $\nu_\mu$ neutrinos.


Introduction
With the recent IceCube observation [1,2] of ultrahigh-energy astrophysical neutrinos, the era of neutrino astronomy has finally begun. The IceCube collaboration has detected a flux of ultra-high-energy cosmic neutrinos (TeV−PeV), which have 5.7σ significance above the atmospheric neutrino backgrounds [2] and thus point to their extraterrestrial origin. Contrary to charged particles which would deflect in magnetic fields in space, such astrophysical neutrinos are expected to point straight back to their sources. The potential impact of understanding these neutrinos ranges from acceleration mechanisms of cosmic rays to fundamental particle physics [3]. Studying the flavor composition of astrophysical neutrinos provides an invaluable tool for exploring these issues. The developments of neutrino telescopes (such as IceCube and alike) [1,2,4] in recent years have stimulated extensive studies [5,6] on the flavor ratios. Given these dedicated studies, it is desirable to find general constraints on the flavor ratios.
In this work, we will derive such general constraints by imposing the unitarity of leptonic mixing matrix [7], because the leptonic mixing modifies the neutrino flavor ratios during their trip from source to detector. The general bounds we obtain do not depend on the neutrino mixing parameters or any experimental input. Especially, we will use leptonic unitarity triangles (LUTs) [8,9] as geometrical means to derive such universal constraints, which turn out to be highly nontrivial. The unitarity bounds are important, because any violation of these bounds would call for new physics, such as activesterile neutrino mixing, neutrino decays, pseudo-Dirac neutrinos, or other exotic effects [10].
We will then apply our general unitarity bounds to the commonly considered sources of ultra-high-energy astrophysical neutrinos, including Pion Sources, Muon-Damped Sources, and Neutron Beam Sources. We compare these bounds with the IceCube measurement [2] and the current global fit of neutrino oscillations [11,12]. From the latest IceCube data [2], we may extract a ratio of muon events to all events of 7/35 = 0.2 , which should equal the ν µ flux ratio. If this ratio will be further confirmed by more dedicated analyses of the collaboration, we show that our unitarity bounds already exclude Pion Sources or Muon-Damped Sources. As another outcome, we will prove that for any astrophysical sources without ν τ neutrinos, if the detected ν µ neutrinos have a flux ratio T < 1/4 , then the source must generate more ν e neutrinos than ν µ neutrinos. These results demonstrate the importance of our general unitarity constraints.
The paper is organized as follows. In Section 2, we will connect the neutrino flavor ratios to the geometrical parameters of the LUTs. Then, we will use our geometrical formulation to analyze the general unitarity constraints on the flavor transition probabilities in Section 3. We apply these constraints to derive nontrivial bounds on the flavor ratios for typical astrophysical neutrino sources, and compare them with the IceCube data in Section 4. Finally, we conclude in Section 5.

Connecting Astrophysical Neutrinos to Leptonic Unitarity Triangles
The leptonic mixing in charged currents is described by the 3×3 unitary matrix U of Pontecorvo-Maki-Nakagawa-Sakata (PMNS) [7]. The orthogonality between the rows (columns) of U forms the LUTs. Following the conventions of our recent study [8], we define the lengths of the three sides of the LUTs, where the subscripts for each length parameter among (a, b, c) are suppressed for simplicity. The flavor transition probability for astrophysical neutrinos ν → ν is given by, which does not contain the oscillation terms since such terms are simply averaged out due to the very large L/E arXiv:1407.3736v1 [hep-ph] 14 Jul 2014 of astrophysical neutrinos. From this, we can further express the transition probability (2) in terms of the LUT parameters, For = , we can classify the flavor appearance probability P → into three cases, Hence, we can rewrite (2) in a matrix form, where the diagonal elements correspond to survival probability, because the full transition probability equals one. For an initial flux from a remote astrophysical neutrino source, let us denote its initial flavor compositions as (Φ e0 , Φ µ0 , Φ τ 0 ). Thus, the detected neutrino flux (after traveling an astronomical distance) can be computed in the matrix form, For neutrino telescopes such as IceCube, the high energy muon neutrinos are in principle distinctive from ν e and ν τ signals as they produce clear muon tracks in the detector. Hence, the flavor ratio Φ µ /Φ tot is a good observable for these experiments [1,2,4], where Φ tot = Φ e + Φ µ + Φ τ . The other possibly measurable ratio is Φ e /Φ tot if the electron neutrino signals can be recognized in the near future. (The flavor ratio Φ τ /Φ tot for tau neutrinos can be deduced from the other two ratios.) The ν µ and ν e flavor ratios are conventionally defined as and thus Φ τ /Φ tot = 1 − T − S . In the literature, sometimes another flavor ratio R ≡ Φ e /Φ τ is introduced to replace S . But the description by (T, S) is equivalent to that of (T, R) because Inspecting the formulas (4), we find that under the exchange ν e ↔ ν µ , the transition probabilities (X, Y, Z) change as follows: X ↔ Y and Z ↔ Z . This also corresponds to the exchanges of the first and second rows (columns) of the matrix P in Eq. (5). With Eqs. (7) and (8), we further infer T ↔ S under the same exchange of ν e ↔ ν µ .
• The πS sources produce neutrinos from pion decays, π → µ + ν µ → e + ν e + 2ν µ , where we do not distinguish the notations between particles and anti-particles for simplicity. Hence, the initial flavor composition is (1 : 2 : 0) . From Eq. (7), the ν µ and ν e flux ratios in this case are given by • The µDS sources produce muon neutrinos in π → µ + ν µ , where the damped muons lose energy so that the neutrino flux produced from their decays is depleted at energies of interest. Hence, the initial flavor composition is (0 : 1 : 0). From Eq. (7), we have the ν µ and ν e flux ratios, • The Neutron Beam Sources (nBS) produce electron neutrinos in beta decay of neutrons. Thus, its initial flavor composition is (1 : 0 : 0). From Eq. (7), the ν µ and ν e flux ratios are given by For the current experiments, which source the detected high-energy astrophysical neutrinos originate from is uncertain. Nevertheless, we note that if all three types of sources are involved, the initial neutrino flux would contain no ν τ neutrinos. Let us consider a general source with mixture [5] from all three types of sources above. In this case, the initial flavor composition can be written as, (η : 1−η : 0), with the parameter η ∈ [0, 1] . Hence, in the general case, we have T and S flux ratios depending on η ,

Unitarity Constraints on Flavor Transitions of Astrophysical Neutrinos
The leptonic mixing matrix of PMNS [7] is unitary, U U † = I , which imposes two kinds of constraints on the row vectors (U 1 , U 2 , U 3 ). These include, (i) the normalization conditions, and (ii) the orthogonal conditions, The second constraint (15) implies the closure of the corresponding unitarity triangle, since the three complex numbers can be represented by three vectors in the complex plane and the zero sum makes them form a closed triangle. In terms of the lengths of three sides (a, b, c), the closure imposes nontrivial triangular inequalities, stating that the sum of the lengths of any two sides is larger than the remaining side, where the equality sign corresponds to the collapse of the triangle into a line. Another equivalent statement is that the difference between the lengths of any two sides is smaller than the remaining side, because a − b c is just b + c a , and so on. Hence, Eq. (16) is sufficient to describe the triangular closure constraints. The geometrical meaning of the first constraint (14) does not appear so obvious, but in fact it restricts the length scale of the three sides. Let us define the notations, (a 1 , b 1 , c 1 ) ≡ (|U 1 |, |U 2 |, |U 3 |) and (a 2 , b 2 , c 2 ) ≡ (|U 1 |, |U 2 |, |U 3 |). Thus, we can express the three sides, (a, b, c) = (a 1 a 2 , b 1 b 2 , c 1 c 2 ). Using the Cauchy-Schwarz inequality [13], we deduce From this, we deduce that the perimeter of the triangle (the sum of its three sides) cannot exceed one, With this, we can further derive an upper bound on the Jarlskog invariant J = Im{U j U * j U k U * k } , with = and j = k [14], fully from geometry. The Euclidean geometry tells us that a shape with fixed perimeter reaches the maximal area when it is a circle, and for a triangle with fixed perimeter, its maximal area is realized when it is a equilateral triangle, with a = b = c and the corresponding area S max = √ 3a 2 /4. (Intuitively, the equilateral triangle in some sense looks like a circle more than any other triangles.) Since the Jarlskog invariant equals twice the area of the LUT, the maximum |J| is given by the equilateral unitarity triangle, |J| max = √ 3a 2 /2 with a = 1/3 . Hence, without using any parameter from the conventional PMNS matrix, we can derive the general geometrical upper bound on |J| , Even though the condition (18) appears quite different from (16), geometrically they are very similar as Fig. 1 illustrates. Each side of the cube in Fig. 1 has length equal 1/2 . Hence, the equation of the plane ABC is, a + b + c = 1 . The inequality (18) is derived from the normalization condition (14), and it requires that the allowed region should be on one side of the plane ABC.
respectively. The unitarity requires (a, b, c) to be a point inside the tetrahedron OABC. and c + a = b , respectively. These planes make a tetrahedron with each side of length 1/ √ 2 . The inequalities (18) and (16) only require that (a, b, c) is a point inside the tetrahedron. Thus, we can immediately infer the upper bound on the length of each side for any LUT, Next, we would ask: what are the unitarity bounds on the averaged transition probabilities (X, Y, Z) defined in Eq. (4)? Here, we can deduce and visualize the bounds geometrically. Consider a sphere with its center at (0, 0, 0). The sphere retains some of the allowed points (a, b, c) on it, and has intersections with the tetrahedron. It should have a radius no larger than 1/ √ 2 . Hence, we deduce Using Eq. (4), we infer the nontrivial upper bound, We stress that we derived these constraints only from the unitarity of the PMNS matrix, without any experimental input. This means that for astrophysical neutrinos (or any neutrinos traveling with a large enough L/E ), the flavor appearance probability for any two flavors ( → ) cannot exceed 1/2 , Another nontrivial result we will prove is that the survival probability is bounded from below, always no smaller than 1/3 , The survival probability P → is just the diagonal elements of the matrix (5). To prove (24), we first choose = e for definiteness, where the terms a 2 τ e + a 2 eµ , for example, can be written as We can derive similar formulas for b 2 τ e +b 2 eµ and c 2 τ e +c 2 eµ . With these, we arrive at This leads to P e→e 1 3 . The first inequality in the second line of Eq. (26) is based on the fact that the arithmetic mean of several real numbers is always smaller than their quadratic mean [13]. Likewise, we can prove that X +Y and Z +X obey the same inequality, With these, we complete the proof of the lower bound (24) on the survival probability. Furthermore, we will prove the following nontrivial inequalities, 2Y +Z, 2Z +X, 2X +Y 25 24 .
We present the proof as follows. Without losing generality, we take Y + 2Z for instance. Let us inspect the difference, Our proof will be achieved so long as we demonstrate the maximum value, G max = 1 24 . Since G only depends on the first two rows of the PMNS matrix, we can generally write the squared elements in a matrix form, where "×" denote elements of no interest here. The quantity G is a function of (x, y, z, w), which may be regarded as equivalent to the four independent parameters of the PMNS matrix. Using the notation (30), we can rewrite the function G , If we overlook the boundary of parameter space, we would naively seek the maximum by solving ∂ This gives a unique solution, x = y = z = w = 1 3 , which results in G = 0 . But, as can be readily checked, this solution is actually a saddle point, rather than the maximum. This implies that the maximum of G should be on the boundary, since this saddle point is the only place where the first derivatives of G vanish. Hence, we will inspect the maximum of G on the boundary of parameter space.
The relevant parameter space is where (x, y, z, w) satisfies (i) x, y, z, w 0 and x + y, z + w 1 , . Any (x, y, z, w) satisfying these two conditions can realize a unitary PMNS matrix. When we are on the boundary of the condition (i), then the first two rows of (30) must contain one zero element. We will prove that only when the second row has a zero element, G realizes its global maximum. In this case, without losing generality, we set the third element of this row be zero, i.e., 1 − z − w = 0 , then we can resolve ∂ x,y,z G| w=1−z = 0 . We find the solution, (x, y, z) = ( 5 12 , 5 12 , 1 2 ) and w = 1 − z = 1 2 . This gives the maximum, Next, we will prove that the other cases either have no extremum or have the extremum not as a global maximum. If the maximum of G is on the boundary of the condition (i), but with the zero element in the first row of (30), we may set 1 − x − y = 0 without losing generality. In this case, we find that the extremum equation ∂ x,z,w G| x=1−y = 0 has no solution by direct calculation.
If the maximum is instead on the boundary of the condition (ii), we have one of the triangle inequalities saturated. Since it is not on the boundary of the condition (i), all the elements of the two rows are non-zero, which means that (a, b, c) are all non-zero. Hence, only one of the triangle inequalities can be saturated. Without losing generality, we consider the situation a + b = c . This is a hypersurface F (x, y, z, w) = 0 in the parameter space where The extremum point can be found by the method of Lagrange multipliers. That is solving ∂ x,y,z,w (G + λF ) = 0 and F = 0 as five equations for (x, y, z, w, λ) . The function G constrained on the hypersurface reaches an extremum with, (x, y, z, w) = ( 7 24 , 7 24 , 5 24 , 5 24 ) and λ = 1 8 . At this point, we find G = 1 48 , which is less than (32). Hence, it is not the global maximum. This completes our proof of (32) and thus the bounds (28).
The inequalities (22), (27) and (28) impose nontrivial unitarity bounds on the transition probabilities (X, Y, Z). We present these bounds in Fig. 2, where the allowed region is surrounded by the colored surfaces. First, the conditions of (22) restrict (X, Y, Z) into a cube (yellow color) with each side length equal to 1 2 . Second, the inequalities of (27) constrain the region through the green planes. Finally, Eq. (28) further bounds the allowed region through the blue planes.

Unitarity Constraints on Flavor Ratios of Astrophysical Neutrinos
As mentioned earlier, the astrophysical neutrinos may originate from different sources. The commonly considered neutrino sources include Pion Sources (πS), Muon-Damped Sources (µDS), and Neutron Beam Sources (nBS). In this section, we will apply the general unitarity bounds (22), (27) and (28) to derive new constraints on the flavor ratios for different sources of cosmic neutrinos. (1 : 2 : 0) As mentioned in Sec. 2, the Pion Sources have the initial neutrino flavor ratio equal (1 : 2 : 0). Thus, we can deduce the flavor ratios at the detector as in (10), (22) and (28)

Pion Sources with Flavor Ratio
Next, we will analyze the unitarity bounds for S +T , S−T , T +2S, T +4S and 3S−T . We may first compute the combinations of T and S , . From the conditions (22) and (28), we infer the unitarity bounds on S +T , The flavor ratio difference S −T in (35) contains 2X + 2Z−Y , which is larger than −Y and smaller than 2(X+ Z) . Thus, from (27) and (28) we derive, which leads to the bound, We note that T + 2S contains the combination 2X + 2Y − Z, which subjects to the same bounds as in (37). Hence, we arrive at The upper and lower bounds on 2X + 4Y − 3Z or 2X + 4Z − 3Y are 25 12 and − 3 2 , respectively, which can be inferred in a similar way to (37). Hence, we can deduce We summarize the above unitarity bounds (34), (36) and (38)-(41) in Table I, for Pion Sources with the initial flavor ratio (1 : 2 : 0). Combining all these constraints, we identify the allowed region of (T, S) in Fig. 3 which  (Table I ) derived from the unitarity of the PMNS matrix without experimental input. These bounds are combined to give the allowed region (light blue area). The dark spot inside the shaded area is a collection of 1000 random points given by the results of a neutrino global fit [11] of the PMNS matrix. The red solid line T = 7/35 (with estimated error band ∆T ±0.083) is the ratio of track/shower events from the recent IceCube data [2], which violates the unitarity bound. The red point depicts (T, S) = ( 1 3 , 1 3 ), and corresponds to a flux ratio of (1 : 1 : 1) at the detector.
is within the shaded area (light blue). In Fig. 3, we also present the parameter region (black points) allowed by the current neutrino global fit [11], The global fit (42) is for the normal mass-ordering. As we have checked, the global fit for inverted mass-ordering only differs a little, and does not lead to any visible effect in our numerical analyses. Thus, it suffices to use the above fit (42) for our present study. Using the global fit (42) for the PMNS parameters (s 13 , s 23 , s 12 , δ D ) with Gaussian distributions, we have generated 1000 random points in Fig. 3. From this plot, we see that these black points appear nearly as a dark spot in the small region of the T − S plane, as required by the current neutrino global fit.
Recently, the IceCube collaboration [2] published 37 candidate events after analyzing its three-year data collection (988 days between 2010 -2013), with deposited energies within the range of 30 -2000 TeV. Among these  (Table I ) derived from the unitarity of the PMNS matrix without experimental input. These bounds are combined to give the allowed region (light blue area). The dark spot inside the shaded area is a collection of 1000 random points given by the result from a neutrino global fit [11] of the PMNS matrix. The red solid line T = 7/35 (with estimated error band ∆T ±0.083) is the ratio of track/shower events from the recent IceCube data [2], which violates the unitarity bound. The red point is defined in the caption of Fig. 3. events 35 are identified as shower events and 7 as muontrack events. IceCube also found [2] that among the 37 recorded events, two events had coincident hits in the IceTop surface array, so they were almost certainly produced in cosmic ray air showers and thus should be subtracted. Although the expected atmospheric background rates have some uncertainty (e.g., from high-mass mesons with shorter lifetimes), the energy spectrum, zenith distribution, and muon track to shower ratio of the observed events strongly disagree with the possibility of having these events from purely atmospheric origin, at 5.7σ level. Hence, these signals should arise from the astrophysical neutrinos with very large L/E, and we have the ν µ flux ratio [2], We may estimate the pure statistical errors of the track events and shower events as √ 7 and √ 35 , respectively. Thus, the flux ratio T has a statistical error of ∆T ±0.083 . We stress that a precise determination of the flux ratio should be eventually done by the experimental collaboration itself. In addition, as the number of signal events is still small, the value of T is likely to be subject to changes after more upcoming data are analyzed. At the current stage, we extract the ratio (43) from the available data [2] and compare it with our gen-eral unitarity bounds.
We present (43) in Fig. 3 by the red solid line (with an error band ∆T ±0.083 ). As we see, if we take Pion Sources with initial flavor ratio (1 : 2 : 0), the ν µ flux ratio T as measured by IceCube is already excluded by the unitarity bounds and the current neutrino global fit [11]. Hence, if future experiments (including IceCube) could further pin down such a small T (lower than 23/72 0.32 ) and confirm the source as Pion Source, then new physics would be required to explain such a small ratio T , such as sterile neutrinos, neutrino decays, pseudo-Dirac neutrinos, or other exotic effects [10]. The comparison with IceCube in Fig. 3 is instructive. It shows that imposing the unitarity bounds can put nontrivial universal constraints on the flux ratios (T, S). It further encourages more elaborated experimental analysis of the IceCube data including all the potential backgrounds and systematics.
In Fig. 3, the dark spot (consisting of the simulated scattered points) gives the region allowed by the global fit of current neutrino data [11]. We note that it almost saturates the unitarity bound on the lower left-hand-side of T , i.e., very close to the unitarity bounds T 23/72 and T + S 47/72 . This shows that these two unitarity bounds are very important.
In passing, we also mark a red point for (T, S) = ( 1 3 , 1 3 ) in Fig. 3, which corresponds to the flux ratio of (1 : 1 : 1) at the detector [15]. This point was recently discussed in [1] [16] for fitting the IceCube data. From  Fig. 3, we see that the red point is excluded by the current neutrino global fit, and lies nearby by the boundary of our unitarity bounds T + S 47/72 0.65 and T 23/72 0.32 . Our finding is consistent with a recent elaborated Monte Carlo analysis [16] (which took into account an energy spectrum and cross sections, etc).  Muon-Damped Sources (µDS) have an initial flavor ratio (0 : 1 : 0). Thus, we can infer the flavor ratios at the detector as in (11), T = 1 − X − Z and S = Z . Using the unitarity conditions (22) and (27), we deduce the bounds, Similar to Sec. 4.1, for the combinations, we derive the following bounds on the flavor ratios, We summarize the above unitarity bounds in Table I for µDS Sources with initial flavor ratio (0 : 1 : 0). We combine these bounds in Fig. 4, and deduce the allowed region of (T, S) which is within the shaded area (light blue). The parameter region allowed by the current neutrino global fit is shown by the black points, which nearly form a dark spot, same as in Fig. 3. We note that in this case the dark spot region almost saturate the unitarity bounds on T from its lower side. This shows that the general bounds T 1 3 and 2T + S 23 24 play an important role here. Furthermore, from Fig. 4 we see that the current IceCube measurement of T (red solid line) violates the unitarity bound if we take µDS as the neutrino sources with initial flavor composition (0 : 1 : 0). Hence, from the general unitarity constraints of Fig. 3−4, we see that the two sources (πS and µDS) of astrophysical neutrinos could not provide the origin of the flux ratio T = 7/35 as extracted from the IceCube data [2].
In the case of nBS sources, we have T = Z and S = 1 − Z − Y . Thus, we derive the combinations S + T , T + 2S, and S − T in terms of (X, Y, Z) , In parallel to Sec. 4.2, we derive unitarity bounds on the flavor ratios, and their combinations above, We summarize these bounds into Table I, and present their combined bounds in Fig. 5. From this plot, it is interesting to see that the IceCube value of T = 7/35 is consistent with the unitarity bounds for the nBS sources. It is also well compatible with the result deduced from the current neutrino global fit [11] (which is represented by the black points in the dark spot region). Comparing with Fig. 3 and Fig. 4, we find that there should be nBS sources with the initial flavor composition (1 : 0 : 0) to serve as the origin of the cosmic neutrinos detected by IceCube. At least, it implies that the fraction of the nBS sources for astrophysical neutrinos should be significant. In the next subsection, we will further analyze a mixed source of the three types, with a generic flux ratio (η : 1−η : 0).

Mixed Sources with Flavor Ratio (η : 1−η : 0)
In general, we can consider a mixed neutrino source of the three types above, where the ν τ neutrinos are absent. So, this general source has the initial flavor ratio (η : 1 − η : 0) with η ∈ [0, 1]. In this notation, Pion Sources correspond to η = 1 3 , Neutron Beam Sources to η = 1 , and Muon-Damped Sources to η = 0 . Thus, for The black straight lines represent the general bounds (Table I ) derived from the unitarity of PMNS matrix without experimental input. These bounds are combined to give the allowed region as shown by the shaded area (light blue). The dark spot inside the shaded area is a collection of 1000 random points given by the current neutrino global fit [11] of the PMNS matrix. The red solid line T = 7/35 (with estimated error band ∆T ±0.083) is the ratio of track/shower events from recent IceCube data [2], which is consistent with the unitarity bounds and the current global fit [11] of the PMNS matrix. The red point is defined in the caption of Fig. 3.
the general case we have the flavor ratios, As we noted below Eq. (9), under the exchange ν e ↔ ν µ , we have, (X, Z) ↔ (Y, Z) and η ↔ (1−η) . Then, from (50), we see that this exchange leads to S ↔ T . This property will also ensure the unitarity bound (the shaded region with light blue color) in Fig. 6 to be symmetric with respect to the line S = T . For a given η and T , we may view (50a) as a straight line in the X −Z plane of Fig. 2, where the slope is fully determined by η , and T only affects the intercept at X = 0 . In the XY Z coordinate frame of Fig. 2, Eq. (51) describes a plane which is perpendicular to the X−Z plane and intersects with it at the line given by (51). It is clear that for η ∈ [0, 1/2) and as T decreases, the plane (51) increases its intercept at Z axis in Fig. 2. If T decreases to a value such that this plane no longer intersects the space surrounded by the colored surfaces in Fig. 2, i.e., every point in this plane violates the unitarity bound, then this value of T is disallowed by the unitarity. Thus, we can derive a lower bound on T by moving the plane (51) to the "critical position" where it is just going to fully leave the colored surfaces (unitarity bounds) of Fig. 2.
Next, we further analyze the unitarity constraints on T +S . From (50), we have From Eqs. (4) and (22), we have 0 (X, Y ) 1 2 . Thus, for η ∈ [0, 1] , we can deduce where the lower bound is reached for X = Y = 1 2 and the upper bound is saturated if X = Y = 0 .
With the formulas from Eq. (50), we deduce the com-  (Table I ) derived from the unitarity of PMNS matrix without experimental input. These bounds are combined to give the allowed region (light blue). The dark spot inside the shaded area collects 1000 random points given by the current neutrino global fit of PMNS matrix and with a scan of η ∈ [0, 1] . The red solid line T = 7/35 (with estimated error band ∆T ±0.083) is the ratio of track/shower events from recent IceCube data [2], which is consistent with the unitarity bounds and the current global fit [11] of the PMNS matrix. The red point is defined in the caption of Fig. 3.
From the above analysis, we summarize the unitarity constraints (54) and (57) for the generic flavor ratio (η : 1−η : 0) in Table I, which hold for any η ∈ [0, 1] . For demonstration, we further present these general bounds in Fig. 6, where we derive the combined unitary bound in the T − S plane, as depicted by the light blue area. As we expected earlier, this unitarity bound (light blue region) is symmetric respect to the line S = T . We note that this general bound holds for any η value and is weaker than the bounds of Figs. 3−5 (which correspond to specific η values). Actually, each bounded area in Figs. 3−5 is contained as a certain part of the allowed region in Fig. 6.
From the general bounds in Fig. 6, we see that even though the type of the cosmic neutrino sources is unknown a priori, we can still deduce nontrivial unitarity constraints on the flux ratios T and S . This means that for any source among (πS, µDS, nBS) or their general mixture in its initial flavor composition, the flux ratios (T, S) must lie in the shaded region (light blue color) of Fig. 6. Otherwise, the unitarity bounds are violated, which would require proper underlying new physics. For comparison, we present the IceCube result (43) as the red solid line (with an estimated error band) in the same plot. This plot also suggests that further measurements of the ν e flux ratio S will be important for pinning down the initial flavor composition in the source.

Conclusions
Observations of ultra-high-energy astrophysical neutrinos at IceCube [1,2] have marked the exciting start of neutrino astronomy. This may eventually help astronomers to map individual sources of astrophysical neutrinos in the sky, and thus paint a picture of the universe by means of neutrino telescopes.
In this work, we made use of the unitarity of leptonic PMNS mixing matrix, and systematically derived uni-tarity constraints on the flavor composition of astrophysical neutrinos, in comparison with the IceCube data [2] and the current neutrino global fit [11,12]. In Section 2, using the leptonic unitarity triangles (LUTs), we formulated the flavor transition probabilities of astrophysical neutrinos in terms of the geometrical parameters of the LUTs, as given in Eqs. (3)− (6). Then, we expressed the ν µ and ν e flux ratios (T, S) by the LUT parameters in Eqs. (10)−(13) for different neutrino sources and their mixture. In Section 3, we quantitatively derived nontrivial unitarity bounds on the transition probabilities of cosmic neutrinos by using the geometrical conditions (such as the triangular inequalities). These are presented in Eqs. (22)-(24) and Eqs. (27)-(28), as well as Fig. 2. These and other new bounds we derived generally hold for three flavor neutrinos, independent of any experimental input.
In Section 4, we applied these generic unitarity bounds to impose constraints on the flux ratios (T, S) for three types of the neutrino sources (πS, µDS, nBS) and their general mixture. These unitarity constraints are summarized in Table I. In Figs. 3−6, we compared these constraints with the IceCube data [2], as well as the current neutrino global fit [11]. We found that the track/shower event ratio ( T = 7/35 = 0.2 ) as extracted from the three-year data of IceCube would violate unitarity bounds if the sources are πS (Fig. 3) or µDS (Fig. 4). But, for neutrino sources such as the nBS or the mixed sources, we revealed that the IceCube result is consistent with our unitarity bounds and the current neutrino global fit, as shown in Figs. 5−6. Even without specifying the type of sources, the suggested flavor ratio (1 : 1 : 1) at the detector is within and very close to our unitarity bound, but is incompatible with the IceCube data . Furthermore, we proved that for any sources without ν τ neutrinos (such as πS, µDS, nBS or their mixture), a detected ν µ flux ratio T < 1/4 will require the initial flavor composition with more ν e neutrinos than ν µ neutrinos.